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A determinantal approach to Sheffer–Appell polynomials via monomiality principle
J. Math. Anal. Appl. 421 (2015) 806–829
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Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa
A determinantal approach to Sheffer–Appell polynomials via monomiality principle Subuhi Khan ∗ , Mumtaz Riyasat Department of Mathematics, Aligarh Muslim University, Aligarh, India
a r t i c l e
i n f o
Article history: Received 20 April 2014 Available online 28 July 2014 Submitted by M.J. Schlosser Keywords: Sheffer–Appell polynomials Determinantal approach Monomiality principle Sheffer–Bernoulli polynomials Sheffer–Euler polynomials
a b s t r a c t In this article, the Sheffer and Appell polynomials are combined to introduce the family of Sheffer–Appell polynomials by using operational methods. The determinantal definition and other properties of the Sheffer–Appell polynomials are established. As particular cases of these polynomials, the Sheffer–Bernoulli and Sheffer–Euler polynomials are introduced and their determinantal definitions are obtained. The operational correspondence between the Appell and Sheffer–Appell polynomials is used to derive the results for the Sheffer–Appell polynomials. Certain results for the Hermite–Appell and Laguerre–Appell polynomials are also obtained. © 2014 Elsevier Inc. All rights reserved.
1. Introduction and preliminaries One of the important classes of polynomial sequences is the class of Sheffer sequences [27]. The class of Appell polynomial sequences [3], which is a subclass of Sheffer polynomial sequences is equally important. The Appell and Sheffer polynomial sequences arise in numerous problems of applied mathematics, theoretical physics, approximation theory and several other mathematical branches. In the past few decades, there has been a renewed interest in Appell and Sheffer sequences. Properties of the Appell and Sheffer sequences are naturally handled within the framework of modern classical umbral calculus by Roman [26]. We recall that, in 1880, Appell [3] introduced and studied sequences of n-degree polynomials An (x),
n = 0, 1, 2, · · · ,
(1.1)
satisfying the recurrence relation d An (x) = nAn−1 (x), dx
n = 1, 2, · · · .
* Corresponding author. E-mail addresses: [email protected] (S. Khan), [email protected] (M. Riyasat). http://dx.doi.org/10.1016/j.jmaa.2014.07.044 0022-247X/© 2014 Elsevier Inc. All rights reserved.
(1.2)
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In particular, Appell noticed the one-to-one correspondence of the set of such sequences {An (x)}n and the set of numerical sequences {An }n , A0 = 0 given by the explicit representation An (x) = An +
n n An−1 x + An−2 x2 + · · · + A0 xn , 1 2
n = 0, 1, 2, · · · .
(1.3)
The above equation, in particular shows explicitly that for each n ≥ 1, An (x) is completely determined by An−1 (x) and by the choice of the constant of integration An . Appell also provided an alternate general method to determine such sequences of polynomials that satisfy relation (1.2). In fact, given the power series
A(t) = A0 +
∞ t t2 tn tn A 1 + A2 + · · · + A n + · · · = An , 1! 2! n! n! n=0
A0 = 0,
(1.4)
with Ai (i = 0, 1, 2, · · ·) real coefficients, a sequence of polynomials satisfying (1.2) is determined by the power series expansion of the product A(t)ext , that is, A(t)ext = A0 (x) +
t t2 tn A1 (x) + A2 (x) + · · · + An (x) + · · · , 1! 2! n!
or, equivalently A(t)ext =
∞
An (x)
n=0
tn . n!
(1.5)
The function A(t)ext is called the generating function of the sequence of polynomials An (x). Alternatively, the sequence An (x) is Appell for g(t) [26], if and only if ∞ tn 1 xt e = An (x) , g(t) n! n=0
(1.6)
where g(t) =
∞
gn
n=0
tn , n!
g0 = 0.
(1.7)
In view of relations (1.5) and (1.6), we have A(t) =
1 . g(t)
(1.8)
There are several ways to define the Sheffer sequences [27], among which by a generating function and by a differential recurrence relation are most common. A polynomial sequence {sn(x)}∞ n=0 (sn (x) being a polynomial of degree n) is called Sheffer A-type zero [25, p. 222 (Theorem 72)] (which we shall hereafter call Sheffer-type), if sn (x) possesses the exponential generating function of the form A(t)exH(t) =
∞
sn (x)
n=0
where A(t) and H(t) have (at least the formal) expansions:
tn , n!
(1.9)
808
S. Khan, M. Riyasat / J. Math. Anal. Appl. 421 (2015) 806–829 ∞
A(t) =
An
tn , n!
A0 = 0
(1.10a)
Hn
tn , n!
H1 = 0,
(1.10b)
n=0
and ∞
H(t) =
n=1
respectively. We know that the order of a formal power series f (t) =
∞
ak t k ,
ak ∈ C,
k=0
denoted by O(f (t)) is the smallest integer k for which the coefficient of tk does not vanish. If O(f (t)) = 1, then the formal power series f (t) has a compositional inverse f¯(t) satisfying f (f¯(t)) = f¯(f (t)) = t. A series f (t) for which O(f (t)) = 1 is called a delta series. We recall the following result [26, p. 17], which can be viewed as an alternate definition of Sheffer sequences. Let f (t) be a delta series and g(t) be an invertible series of the following forms: f (t) =
∞
fn
n=0
tn , n!
f0 = 0, f1 = 0
(1.11a)
and g(t) =
∞
gn
n=0
tn , n!
g0 = 0.
(1.11b)
Then there exists a unique sequence sn (x) of polynomials satisfying the orthogonality conditions g(t)f (t)k sn (x) = n!δn,k ,
∀n, k ≥ 0.
(1.12)
According to Roman [26, p. 18 (Theorem 2.3.4)], the polynomial sequence sn (x) is uniquely determined by two (formal) power series given by Eqs. (1.11a) and (1.11b). The exponential generating function of sn (x) is then given by 1 g(f −1 (t))
ex(f
−1
(t))
=
∞
sn (x)
n=0
tn , n!
(1.13)
for all x in C, where f −1 (t) is the compositional inverse of f (t). In view of Eqs. (1.9) and (1.13), we have A(t) =
1 g(f −1 (t))
(1.14)
and H(t) = f −1 (t).
(1.15)
The sequence sn (x) in Eq. (1.12) is the Sheffer sequence for the pair (g(t), f (t)). The Sheffer sequence for (1, f (t)) is called the associated Sheffer sequence for f (t) and the Sheffer sequence for (g(t), t) becomes
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the Appell sequence for g(t) [26, p. 17]. Also, from Eqs. (1.9) (or (1.13)) and (1.5) (or (1.6)), it follows that for H(t) = f −1 (t), the Sheffer polynomials sn (x) reduce to the Appell polynomials An (x). ˆ and Pˆ Next, we recall that according to the monomiality principle [28,11], there exist two operators M playing, respectively, the role of multiplicative and derivative operators for a polynomial set {pn (x)}n∈N , ˆ and Pˆ satisfy the following identities, for all n ∈ N: that is, M ˆ pn (x) = pn+1 (x) M
(1.16)
Pˆ pn (x) = npn−1 (x).
(1.17)
and
ˆ and Pˆ must satisfy the The polynomial set {pn (x)} is then called a quasi-monomial. The operators M commutation relation ˆ ] = Pˆ M ˆ −M ˆ Pˆ = ˆ1 [Pˆ , M
(1.18)
and thus display a Weyl group structure. If the considered polynomial set {pn (x)} is quasi-monomial, its properties can be easily derived from ˆ and Pˆ operators. In fact the following hold: each of the M ˆ and Pˆ have differential realizations, then the polynomials pn (x) satisfy the differential equation (i) If M ˆ Pˆ pn (x) = npn (x). M
(1.19)
(ii) Assuming that p0 (x) = 1, then pn (x) can be explicitly constructed as ˆ n {1}. pn (x) = M
(1.20)
(iii) In view of identity (1.20) the exponential generating function of pn (x) can be given in the form ˆ
etM {1} =
∞
pn (x)
n=0
tn , n!
|t| < ∞.
(1.21)
The Appell polynomials An (x) are quasi-monomial with respect to the following multiplicative and derivative operators: ˆ A = x + A (Dx ) , M A(Dx )
(1.22a)
ˆ A = x − g (Dx ) M g(Dx )
(1.22b)
or, equivalently
and PˆA = Dx , respectively.
(1.23)
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The Sheffer polynomials sn (x) and the monomiality principle, along with the underlying operational formalism, offer a powerful tool for investigation of the properties of a wide class of polynomials, see for example [5,13,24]. It has been shown in [24], that if sn (x) are of Sheffer-type then it is possible to give ˆ and Pˆ . Conversely, if M ˆ =M ˆ (x, Dx ) and Pˆ = Pˆ (Dx ), then sn (x) satisfying explicit representations of M Eqs. (1.16) and (1.17) are necessarily of Sheffer-type. The multiplicative and derivative operators (also known as raising and lowering operators) for the Sheffer polynomials sn (x) are given by −1
ˆ s = xH H −1 (Dx ) + A (H (Dx )) , M A(H −1 (Dx ))
(1.24a)
or, equivalently g (Dx ) 1 x− g(Dx ) f (Dx )
ˆs = M
(1.24b)
and Pˆs = H −1 (Dx ),
(1.25a)
Pˆs = f (Dx ),
(1.25b)
or, equivalently
respectively. Costabile et al. [8] has given a new approach to Bernoulli polynomials based on a determinantal definition. This approach is further extended to provide determinantal definitions of the Appell and Sheffer polynomial sequences by Costabile and Longo in [9] and [10] respectively. The equivalence of determinantal approach with other existing approaches is also established. The simplicity of the algebraic approach to the Appell and Sheffer sequences established in [9,10], allows several applications. The above mentioned research works of Costabile and Longo and the importance of operational methods in the theory of special functions motivated the authors to introduce and study the Sheffer–Appell polynomials by using operational techniques and determinantal approach. In this paper, the Sheffer–Appell polynomials are introduced by means of generating function, series definition and determinantal definition. These polynomials are framed within the context of monomiality principle and their properties are derived. The corresponding results for the Sheffer–Bernoulli and Sheffer– Euler polynomials are also obtained. Examples of some members belonging to the family of Sheffer–Appell polynomials are considered and the graphs of certain members are drawn for suitable values of the indices. 2. Sheffer–Appell polynomials The polynomials defined as the discrete convolution of the known polynomials are used to explore new (A) families of special polynomials. The discrete Appell convolution fn (x) is defined as fn(A) (x)
=
n n k=0
k
Ak fn−k (x). (A)
Taking the generic polynomials fn (x) (n ∈ N, x ∈ R) as xn in the above equation, we find that fn (x) reduce to the Appell polynomials An (x) [26]. The Sheffer–Appell polynomials denoted by s An (x) are defined as the discrete Appell convolution of the Sheffer polynomials sn (x).
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In this section, we introduce the Sheffer–Appell polynomials by means of generating function and series definition. Further, a determinantal definition of the Sheffer–Appell polynomials is given. First, we derive the generating function for the Sheffer–Appell polynomials by proving the following result. Theorem 2.1. The generating function for the Sheffer–Appell polynomials s An (x) is given as: A(t)A(t)exH(t) =
∞
s An (x)
n=0
tn , n!
(2.1a)
or, equivalently ∞ 1 x(f −1 (t)) tn e . = s An (x) −1 g(f (t)) g(t) n! n=0
1
(2.1b)
ˆ s of the Sheffer polynomials sn (x) in the l.h.s. of Proof. Replacing x by the multiplicative operator M generating function (1.5) of the Appell polynomials and denoting the resultant Sheffer–Appell polynomials in the r.h.s. by s An (x), we have ˆ
A(t)eMs t {1} =
∞ n=0
s An (x)
tn , n!
(2.2)
which in view of Eq. (1.21) gives A(t)
∞
sn (x)
n=0
∞ tn tn = . s An (x) n! n=0 n!
(2.3)
Using generating function (1.9) of the Sheffer polynomials sn (x) in the l.h.s. of the above equation we get assertion (2.1a). Also, in view of relations (1.8), (1.14) and (1.15), Eq. (2.1a) can be expressed equivalently as (2.1b). 2 Next, we obtain the series definition of the Sheffer–Appell polynomials s An (x) by proving following result. Theorem 2.2. The Sheffer–Appell polynomials s An (x) are defined by the series: s An (x) =
n n Ak sn−k (x). k
(2.4)
k=0
Proof. Using expansion (1.4) of A(t) in the l.h.s. of Eq. (2.3), we find ∞ ∞ n=0 k=0
Ak sn (x)
∞ tn+k tn = . s An (x) n!k! n! n=0
(2.5)
Replacing n by n − k in the l.h.s. of Eq. (2.5) and then equating the coefficients of like powers of t in both sides of the resultant equation, we get assertion (2.4). 2 Note. It is important to note that definition (2.4) is a direct consequence of the fact that Sheffer–Appell polynomials s An (x) are discrete Appell convolution of the Sheffer polynomials sn (x).
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Remark 2.1. From Eqs. (1.5) and (2.2), we get the following operational correspondence between the Sheffer– Appell polynomials s An (x) and Appell polynomials An (x): ˆ s ). = An (M
s An (x)
(2.6)
In order to give the determinantal definition of the Sheffer–Appell polynomials s An (x), we prove the following result. Theorem 2.3. The Sheffer–Appell polynomials s An (x) of degree n are defined by
s A0 (x)
=
1 , β0
β0 =
n
s An (x)
=
(−1)
n+1
(β0 )
1 β 0 0 0 . . 0
1 , A0
(2.7)
s1 (x)
s2 (x)
···
sn−1 (x)
β1
···
β0
β2
2 1 β1
0
β0
···
βn−1
n−1 1 βn−2
n−1 2 βn−3
.
.
···
.
.
.
···
.
0
0
···
β0
···
sn (x) βn
n 1 βn−1
n , 2 βn−2 . .
n n−1 β1
1 βn = − A0
n n k=1
n = 1, 2, 3, · · · ,
k
Ak βn−k ,
(2.8)
where β0 , β1 , · · · , βn ∈ R, β0 = 0 and sn (x) (n = 0, 1, · · ·), are the Sheffer polynomials defined by Eq. (1.9) (or (1.13)). Proof. First, we consider the following determinantal definition of the Appell polynomials An (x) of degree n given by Costabile and Longo [9, p. 1533]:
A0 (x) =
1 , β0
β0 =
n
An (x) =
(−1)
n+1
(β0 )
n = 1, 2, 3, · · · ,
1 β 0 0 0 . . 0
1 , A0
(2.9)
x
x2
···
β1
···
β0
β2
2 1 β1
0
β0
.
.
···
.
.
.
···
.
0
0
···
β0
xn−1
βn−1
n−1 ··· 1 βn−2
n−1 ··· 2 βn−3
βn
n βn−1
n1 , 2 βn−2 . .
n β xn
n−1
n
1 n βn = − Ak βn−k , A0 k k=1
1
(2.10)
where β0 , β1 , · · · , βn ∈ R, β0 = 0. Taking n = 0 in series definition (2.4) of the Sheffer–Appell polynomials and then using Eq. (2.9) in the resultant equation, we get assertion (2.7). In order to prove assertion (2.8), we expand the determinant given in Eq. (2.10) with respect to the first row, so that we have
S. Khan, M. Riyasat / J. Math. Anal. Appl. 421 (2015) 806–829
An (x) =
(−1)n (β0 )n+1
β1 β0 0 . . 0
β2
2 1 β1 β0
···
.
···
.
0
···
β0
+
···
.
.
···
.
0
···
β0
β0 0 0 . . 0
···
0
βn−1
n−1 ··· 1 βn−2
n−1 ··· 2 βn−3
.
···
.
.
···
.
0
···
β0
β0
β0 0 0 . . 0
(−1) x (β0 )n+1
x (β0 )n+1
.
β1
···
β0
β2
2 1 β1
0
β0
···
.
.
···
β1
···
.
.
···
0
0
···
···
β0
β2
2 1 β1
0
β0
···
.
.
···
.
.
···
0
0
···
β1
βn
n 1 βn−1
n 2 βn−2 . .
n n−1 β1
βn−1
n−1 ··· 1 βn−2
n−1 ··· 2 βn−3
β0
2n−1 n−1
.
···
β2
2 1 β1
β0 0 (−1)n x2 0 + (β0 )n+1 . . 0
n
βn−1
n−1 ··· 1 βn−2
n−1 ··· 2 βn−3
.
β0 0 n 0 (−1) x − (β0 )n+1 . . 0
+
···
···
813
βn
n β n−1
n1 2 βn−2 . .
n n−1 β1 βn
n 1 βn−1
n 2 βn−2 + ··· . .
n n−1 β1 βn
n 1 βn−1
n 2 βn−2 . .
n n−1 β1
βn−1
n−1 1 βn−2
n−1 2 βn−3 . . . β0
(2.11)
ˆ s in Eq. (2.11) and then Since each minor in Eq. (2.11) is independent of x, therefore replacing x by M using the fact that s0 (x) = 1 and sn (x) = Msn {s0 (x)} (n = 1, 2, 3, · · ·) in the r.h.s. of the resultant equation, we find
n ˆ s ) = (−1) An (M (β0 )n+1
β1 β0 0 . . 0
β2
2 1 β1 β0
···
βn−1
n−1 ··· 1 βn−2
n−1 ··· 2 βn−3
.
···
.
.
···
.
0
···
β0
βn
n 1 βn−1
n 2 βn−2 . .
n n−1 β1
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S. Khan, M. Riyasat / J. Math. Anal. Appl. 421 (2015) 806–829
β0 0 (−1)n s1 (x) 0 − (β0 )n+1 . . 0 β0 0 n (−1) s2 (x) 0 + (β0 )n+1 . . 0
βn
n 1 βn−1
n β0 2 βn−2 . ··· . . . ··· . .
n 0 ··· β0 n−1 β1 β1 · · · βn−1 βn
n β0 · · · n−1 β β n−2 n−1 1
n−1
n1 0 ··· 2 βn−3 2 βn−2 + ··· . ··· . . . ··· . .
n 0 ··· β0 n−1 β1 β0 β1 β2 ··· βn
2 0 β · · · n1 βn−1 0 1 β1
(−1)2n−1 sn−1 (x) 0 0 β0 · · · n2 βn−2 + (β0 )n+1 . . ··· . . . . . ··· .
n 0 0 0 ··· n−1 β1 β2 ··· βn−1 β0 β1
2 n−1 0 β0 1 β1 · · · 1 βn−2
n−1 0 β0 ··· sn (x) 0 2 βn−3 + . n+1 (β0 ) . . ··· . . . . . · · · .
n 0 0 0 ··· β 1 n−1 β2
2 1 β1
···
βn−1
n−1 ··· 1 βn−2
n−1 ··· 2 βn−3
(2.12)
Now, using operational rule (2.6) in the l.h.s. and combining the terms in the r.h.s. of Eq. (2.12), we get assertion (2.8). 2 The Appell and Sheffer polynomials, An (x) and sn (x) respectively, are quasi-monomial. In order to show that the Sheffer–Appell polynomials s An (x) are quasi-monomial, we prove the following result. Theorem 2.4. The Sheffer–Appell polynomials s An (x) are quasi-monomial with respect to the following multiplicative and derivative operators: −1 −1
ˆ sA = xH H −1 (Dx ) + A (H (Dx )) + A (H (Dx )) , M −1 −1 A(H (Dx )) A(H (Dx ))
(2.13a)
or, equivalently ˆ sA = M
g (f (Dx )) g (Dx ) 1 − x− g(Dx ) f (Dx ) g(f (Dx ))
(2.13b)
and PˆsA = H −1 (Dx ),
(2.14a)
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or, equivalently PˆsA = f (Dx ),
(2.14b)
respectively. Proof. Differentiating Eq. (2.1a) partially with respect to t, we find
xH (t) +
∞ A (t) A (t) tn + A(t)A(t)exH(t) = . s An+1 (x) A(t) A(t) n! n=0
(2.15)
(t) A (t) Since A(t) and A(t) are invertible series of t, therefore A A(t) and A(t) possess power series expansions of t. Thus, in view of the following identity for the Sheffer–Appell polynomials:
Dx A(t)A(t)exH(t) = H(t)A(t)A(t)exH(t) ,
(2.16a)
H −1 (Dx ) A(t)A(t)exH(t) = tA(t)A(t)exH(t) ,
(2.16b)
or, equivalently
Eq. (2.15) becomes
xH H
−1
∞ A (H −1 (Dx )) A (H −1 (Dx )) tn xH(t) + A(t)A(t)e . (Dx ) + = A (x) n+1 s A(H −1 (Dx )) A(H −1 (Dx )) n! n=0
(2.17)
Again, using generating function (2.1a) in the l.h.s. of Eq. (2.17) and rearranging the summation, we have ∞ n=0
A (H −1 (Dx )) A (H −1 (Dx )) xH H −1 (Dx ) + + A(H −1 (Dx )) A(H −1 (Dx ))
s
tn An (x) n!
=
∞
s An+1 (x)
n=0
tn . n!
Equating the coefficients of the same powers of t in both sides of the above equation, we find
A (H −1 (Dx )) A (H −1 (Dx )) s xH H −1 (Dx ) + + An (x) = s An+1 (x), −1 −1 A(H (Dx )) A(H (Dx ))
(2.18)
which in view of monomiality principle equation (1.16) yields assertion (2.13a). Also, in view of relations (1.8), (1.14) and (1.15), Eq. (2.13a) can be expressed equivalently as (2.13b). In order to prove assertion (2.14a), we use generating function (2.1a) in both sides of the identity (2.16b), so that we have ∞ ∞ tn tn −1 = . (2.19) H (Dx ) s An (x) s An−1 (x) n! (n − 1)! n=0 n=1 Rearranging the summation in the l.h.s. of Eq. (2.19) and then equating the coefficients of the same powers of t in both sides of the resultant equation, we find s
H −1 (Dx )
An (x) = ns An−1 (x),
n ≥ 1,
(2.20)
which in view of monomiality principle equation (1.17) yields assertion (2.14a). Also, in view of relation (1.15), Eq. (2.14a) can be expressed equivalently as (2.14b). 2
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S. Khan, M. Riyasat / J. Math. Anal. Appl. 421 (2015) 806–829
Remark 2.2. We remark that Eqs. (2.18) and (2.20) are the differential recurrence relations satisfied by the Sheffer–Appell polynomials s An (x). Theorem 2.5. The Sheffer–Appell polynomials s An (x) satisfy the following differential equation:
−1 −1 A (H −1 (Dx )) −1 A (H −1 (Dx )) −1 xH H (Dx ) H (Dx ) + H (Dx ) + H (Dx ) − n s An (x) = 0, A(H −1 (Dx )) A(H −1 (Dx ))
(2.21a)
or, equivalently g (Dx ) f (Dx ) g (f (Dx )) x− − f (D ) − n x s An (x) = 0. g(Dx ) f (Dx ) g(f (Dx ))
(2.21b)
Proof. Using Eqs. (2.13a) (or (2.13b)) and (2.14a) (or (2.14b)) in monomiality principle equation (1.19), we get assertion (2.21a) (or (2.21b)). 2 Further, we remark that operational rule (2.6) can be used to derive the results for the Sheffer–Appell polynomials from the results of the Appell polynomials. Several identities involving Appell polynomials are known. To give an example, we consider the following results for the Appell polynomials An (x) [9, (31)–(32) p. 1534]:
n−1 n 1 An (x) = xn − βn−k Ak (x) , n = 1, 2, · · · , β0 k k=0 n n n x = βn−k Ak (x), n = 0, 1, · · · . k
(2.22)
(2.23)
k=0
ˆ s in Eqs. (2.22), (2.23) and then using operational rules (2.6) and in view of Eq. (1.20), Replacing x by M we find the following results for the Sheffer–Appell polynomials s An (x):
n−1 n 1 sn (x) − βn−ks Ak (x) , n = 1, 2, · · · , s An (x) = β0 k k=0 n n sn (x) = βn−ks Ak (x), n = 0, 1, · · · . k
(2.24)
(2.25)
k=0
It is important to note that the Sheffer–Appell polynomials introduced in this section are actually the Sheffer polynomials, since their generating function is of the type A(t)exH(t) , with a suitable choice for A (t). In the next section, the Sheffer–Bernoulli and Sheffer–Euler polynomials are introduced as special cases of the Sheffer–Appell polynomials s An (x). 3. Sheffer–Bernoulli and Sheffer–Euler polynomials The typical examples of Appell polynomials besides the trivial example {xn }∞ n=0 are the Bernoulli, Euler and Hermite polynomials. In particular, Hermite polynomials sequence Hen (x) defined by the generating function t2
e(xt− 2 ) =
∞ n=0
Hen (x)
tn , n!
|t| < ∞; |x| < ∞
(3.1)
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817
Table 1 Some known Appell polynomials. S. No.
g(t); A(t) g(t) =
(et −1) ; t
g(t) =
(et +1) ; 2
III.
g(t) =
(e −1) tα
IV.
g(t) =
(et +1)α 2α
V.
g(t) =
I. II.
A(t) =
t
α
A(t) =
t (et −1)
Generating functions xt t tn = ∞ n=0 Bn (x) n! (et −1) e
A(t) =
2 (et +1)
xt 2 (et +1) e
t (et −1)α
t (et −1)α
ext =
; A(t) =
2α (et +1)α
2α (et +1)α
ext =
∞
(eα1 t −1)(eα2 t −1)...(eαm t −1) ; α1 α2 ...αm tm α1 α2 ...αm tm (eα1 t −1)(eα2 t −1)...(eαm t −1) α1 t
=
tn n=0 En (x) n!
; A(t) =
α
α2 t
αm t
VI.
+1) g(t) = (e +1)(e 2+1)...(e ; m m A(t) = (eα1 t +1)(eα2 t2+1)...(eαm t +1)
VII.
g(t) = A(t) =
h et − m−1 (t ) h=0 h! ; tm tm th et − m−1 ( ) h=0
The generalized Bernoulli polynomials [16]
n
n=0
(α) En (x) tn!
The generalized Euler polynomials [16]
α1 α2 ...αm tm xt (eα1 t −1)(eα2 t −1)...(eαm t −1) e ∞ n (m) = n=0 Bn (x|α1 , α2 , · · · , αm ) tn!
The generalized Bernoulli polynomials of order m [15]
xt 2m (eα1 t +1)(eα2 t +1)...(eαm t +1) e ∞ n (m) = n=0 En (x|α1 , α2 , · · · , αm ) tn!
The generalized Euler polynomials of order m [15]
h!
IX.
g(t) =
X.
g(t) = ( λet2+1 )−α ; A(t) = ( λet2+1 )α
XI.
g(t) =
XII.
g(t) = e−(ξ◦ +ξ1 t+ξ2 t +···+ξr+1 t ) ; 2 r+1 A(t) = e(ξ◦ +ξ1 t+ξ2 t +···+ξr+1 t )
=
∞ n=0
XIII.
g(t) = (1 − t)m+1 ; A(t) =
XIV.
g(t) =
∞
( λett−1 )α ext =
A(t) =
A(t) =
A(t) =
xt t λ et −1 e
t λet −1
2 λet +1
2
(et +1) ; 2t
(α) Bn (x) tn!
n
[m−1] Bn (x) tn!
h!
g(t) = ( λett−1 )−α ; A(t) = ( λett−1 )α
λet +1 ; 2
The Euler polynomials [25] n
n=0
tm ext h et − m−1 (t ) h=0
VIII.
λet −1 ; t
The Bernoulli polynomials [25]
∞
∞
α
Polynomials
r+1
1 (1−t)m+1
2t (et +1)
=
n
n=0
∞
t B(α) n (x; λ) n!
n
Bn (x; λ) tn!
n=0
(α) tn ( λet2+1 )α ext = ∞ n=0 En (x; λ) n! α |t + log λ| < π; 1 := 1 xt 2 tn = ∞ n=0 En (x; λ) n! λet +1 e |t + log λ| < π 2
1 t+ξ2 t +···+ξr+1 t e(ξ◦ +ξ tn = ∞ n=0 An (x) n!
1 (1−t)m+1
ext =
xt 2t (et +1) e
=
∞ n=0
∞ n=0
r+1
) xt
e
G(m) (x)tn , n
n
Gn (x) tn! ,
The new generalized Bernoulli polynomials [7] The Apostol–Bernoulli polynomials of order α [22] The Apostol–Bernoulli polynomials [2,22] The Apostol–Euler polynomials of order α [21,22] The Apostol–Euler polynomials [2,21,22] The generalized Gould–Hopper polynomials [14] (For r = 1, the Hermite polynomials Hn (x) [1] and for r = 2, classical 2-orthogonal polynomials) The Miller–Lee polynomials [1,12] (For m = 0, the truncated exponential polynomials en (x) [1] and for m = β − 1, the modified Laguerre polynomials (β) fn (x) [23]) The Genocchi polynomials [13]
is a unique sequence of Appell polynomials that is also orthogonal with respect to a positive measure. Some known Appell polynomials are listed in Table 1. The Bernoulli polynomials Bn (x) and Euler polynomials En (x) are important members of the Appell family. Therefore, in this section, we introduce the Sheffer–Bernoulli and Sheffer–Euler polynomials, s Bn (x) and s En (x) respectively, by means of generating functions and series definitions. The determinantal definitions for these polynomials are also given. 1 t Taking A(t) (or g(t) ) = et −1 of the Bernoulli polynomials Bn (x) in the l.h.s. of generating function (2.1a) (or (2.1b)) and denoting the resultant Sheffer–Bernoulli polynomials in the r.h.s. by sBn (x), we get the following generating function for the Sheffer–Bernoulli polynomials s Bn (x): ∞ tn t xH(t) A(t)e , = s Bn (x) t e −1 n! n=0
(3.2a)
818
S. Khan, M. Riyasat / J. Math. Anal. Appl. 421 (2015) 806–829
or, equivalently ∞ 1 tn t x(f −1 (t)) e . = s Bn (x) t −1 e − 1 g(f (t)) n! n=0
(3.2b)
1 Similarly, taking A(t) (or g(t) ) = et2+1 of the Euler polynomials En (x) in the l.h.s. of generating function (2.1a) (or (2.1b)) and denoting the resultant Sheffer–Euler polynomials in the r.h.s. by s En (x), we get the following generating function for the Sheffer–Euler polynomials s En (x): ∞ tn 2 xH(t) A(t)e , = E (x) n s et + 1 n! n=0
(3.3a)
∞ 1 tn 2 x(f −1 (t)) e . = E (x) n s et + 1 g(f −1 (t)) n! n=0
(3.3b)
or, equivalently
Now, taking An (x) = Bn (x) in the l.h.s. and Ak = Bk in the r.h.s. of Eq. (2.4), we get the following series definition for the Sheffer–Bernoulli polynomials s Bn (x):
s Bn (x) =
n n Bk sn−k (x). k
(3.4)
k=0
Similarly, for An (x) = En (x) and Ak = Ek , we get the following series definition for the Sheffer–Euler polynomials s En (x):
s En (x) =
n n Ek sn−k (x). k
(3.5)
k=0
Remark 3.1. Next, taking An (x) = Bn (x) in Eq. (2.6), we get the following relation between the Bernoulli and Sheffer–Bernoulli polynomials: s Bn (x)
ˆ s ). = Bn (M
(3.6)
Similarly, for An (x) = En (x), we get the following relation between the Euler and Sheffer–Euler polynomials: s En (x)
ˆ s ). = En (M
(3.7)
1 It has been shown in [9] that for β0 = 1 and βi = i+1 (i = 1, 2, · · · , n) the determinantal definition of the Appell polynomials An (x) given by Eqs. (2.9) and (2.10) reduces to the determinantal form of the Bernoulli polynomials Bn (x) [8]. 1 Therefore, taking β0 = 1 and βi = i+1 (i = 1, 2, · · · , n) in Eqs. (2.7) and (2.8), we get the following determinantal definition of the Sheffer–Bernoulli polynomials s Bn (x):
S. Khan, M. Riyasat / J. Math. Anal. Appl. 421 (2015) 806–829
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Definition 3.1. The Sheffer–Bernoulli polynomials s Bn (x) of degree n are defined by s B0 (x)
= 1,
1 s1 (x) s2 (x) 1 1 1 2
23 1 0 1 1 2 n 0 1 s Bn (x) = (−1) 0 . . . . . . 0 0 0
··· ··· ··· ···
sn−1 (x) 1
n−1n
1
1
n−1 n−1 1 2
n−2
···
.
···
.
···
1
sn (x) 1
n+1 n 1 1 n
n 1 , 2 n−1 . .
n 1
(3.8)
n = 1, 2, · · · ,
(3.9)
n−1 2
where sn (x) (n = 0, 1, 2, · · ·) are the Sheffer polynomials of degree n. Further, in view of the fact that for β0 = 1 and βi = 12 (i = 1, 2, · · · , n), Eqs. (2.9) and (2.10) give the determinantal form of the Euler polynomials [9, (60)–(61) p. 1540], so we get the following determinantal definition of the Sheffer–Euler polynomials s En (x): Definition 3.2. The Sheffer–Euler polynomials s En (x) of degree n are defined by s E0 (x)
= 1,
1 s1 (x) 1 1 2 0 1 n 0 s En (x) = (−1) 0 . . . . 0 0
s2 (x) · · · sn−1 (x) 1
2 1 2
···
1
2 1 n−1
2 1
···
1
···
2
1 1 n−1 2 2
.
···
.
.
···
.
0
···
1
sn (x) 1 2
1 n 2 1
1 n , 2 2 . .
n 1
(3.10)
n = 1, 2, · · · ,
(3.11)
2 n−1
where sn (x) (n = 0, 1, 2, · · ·) are the Sheffer polynomials of degree n. In order to frame the Sheffer–Bernoulli and Sheffer–Euler polynomials, sBn (x) and s En (x) respectively, within the context of monomiality principle, we prove the following results: Theorem 3.1. The Sheffer–Bernoulli polynomials s Bn (x) are quasi-monomial with respect to the following multiplicative and derivative operators: −1
−1 (H (Dx ))
(1 − (H −1 (Dx ))) − 1 ˆ sB = xH H −1 (Dx ) + A (H (Dx )) + e , M A(H −1 (Dx )) H −1 (Dx )(e(H −1 (Dx )) − 1)
(3.12a)
or, equivalently ˆ sB = M
1 ef (Dx ) (1 − f (Dx )) − 1 g (Dx ) + x− g(Dx ) f (Dx ) f (Dx )(ef (Dx ) − 1)
(3.12b)
and PˆsB = H −1 (Dx ),
(3.13a)
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820
or, equivalently PˆsB = f (Dx ),
(3.13b)
respectively. Proof. For the Bernoulli polynomials Bn (x), we have A(t) (or −1
(H (Dx )) e(H −1 (Dx )) −1
and consequently A (H −1 (Dx )) =
e
(H −1 (Dx ))
1 t g(t) ) = et −1 .
Therefore, taking A(H −1 (Dx )) =
−1
(1−(H (Dx )))−1 (e(H −1 (Dx )) −1)2
in Eq. (2.13a), we get assertion
(3.12a). (f (Dx ))
(f (Dx )−1)+1 Similarly, taking g(f (Dx )) = e f (Dx−1 and consequently g (f (Dx )) = e in Eq. (2.13b), ) (f (Dx ))2 we get assertion (3.12b). Further, in view of Eq. (2.14a) (or (2.14b)), we get assertion (3.13a) (or (3.13b)). 2 f (Dx )
Theorem 3.2. The Sheffer–Bernoulli polynomials s Bn (x) satisfy the following differential equation:
A (H −1 (Dx )) −1 xH H −1 (Dx ) H −1 (Dx ) + H (Dx ) A(H −1 (Dx )) −1 e(H (Dx )) (1 − (H −1 (Dx ))) − 1 −1 + H (Dx ) − n s Bn (x) = 0, H −1 (Dx )(e(H −1 (Dx )) − 1)
(3.14a)
or, equivalently g (Dx ) f (Dx ) ef (Dx ) (1 − f (Dx )) − 1 f (Dx ) − n s Bn (x) = 0. x− + g(Dx ) f (Dx ) f (Dx )(ef (Dx ) − 1)
(3.14b)
Proof. Using Eqs. (3.12a) (or (3.12b)) and (3.13a) (or (3.13b)) in monomiality principle equation (1.19), we get assertion (3.14a) (or (3.14b)). 2 Theorem 3.3. The Sheffer–Euler polynomials s En (x) are quasi-monomial with respect to the following multiplicative and derivative operators: −1
−1 (H (Dx ))
ˆ sE = xH H −1 (Dx ) + A (H (Dx )) − e −1 , M −1 H (Dx ) + 1) A(H (Dx )) (e
(3.15a)
or, equivalently ˆ sE = M
x−
1 ef (Dx ) g (Dx ) − f (D ) g(Dx ) f (Dx ) (e x + 1)
(3.15b)
PˆsE = H −1 (Dx ),
(3.16a)
PˆsB = f (Dx ),
(3.16b)
and
or, equivalently
respectively.
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Proof. For the Euler polynomials En (x), we have A(t) (or 2 e(H −1 (Dx )) +1
and consequently A (H −1 (Dx )) = −
1 g(t) )
−1 2e(H (Dx )) (e(H −1 (Dx )) +1)2
=
2 et +1 .
821
Therefore, taking A(H −1 (Dx )) =
in Eq. (2.13a), we get assertion (3.15a).
f (Dx )
f (Dx )
Similarly, taking g(f (Dx )) = e 2 +1 and consequently g (f (Dx )) = e 2 in Eq. (2.13b), we get assertion (3.15b). Further, in view of Eq. (2.14a) (or (2.14b)), we get assertion (3.16a) (or (3.16b)). 2 Theorem 3.4. The Sheffer–Euler polynomials s En (x) satisfy the following differential equation:
−1
A (H −1 (Dx )) −1 e(H (Dx )) −1 xH H −1 (Dx ) H −1 (Dx ) + H H (D ) − (D ) − n x x s En (x) = 0, A(H −1 (Dx )) (eH −1 (Dx ) + 1) (3.17a)
or, equivalently g (Dx ) f (Dx ) ef (Dx ) − f (Dx ) − n s En (x) = 0. x− g(Dx ) f (Dx ) ef (Dx ) + 1
(3.17b)
Proof. Using Eqs. (3.15a) (or (3.15b)) and (3.16a) (or (3.16b)) in monomiality principle equation (1.19), we get assertion (3.17a) (or (3.17b)). 2 Next, we consider the applications of operational rules (3.6) and (3.7) to derive the results for the Sheffer–Bernoulli and the Sheffer–Euler polynomials, from the results of the Bernoulli and Euler polynomials respectively. We recall the following functional equations involving the Bernoulli and Euler polynomials: n−1 k=0
n Bk (x) = nxn−1 , k
n = 2, 3, · · · ,
n−1 1 n En (x) = x − n Ek (x), 2 k n
n = 1, 2, · · · .
(3.18)
(3.19)
k=0
ˆ s of the Sheffer polynomials sn (x) and then using apReplacing x by the multiplicative operator M propriate operational rules in the resultant equations, we find the following functional equations for the Sheffer–Bernoulli and Sheffer–Euler polynomials: n−1 k=0
n s Bk (x) = nsn−1 (x), k
n−1 1 n s En (x) = sn (x) − n s Ek (x), 2 k
n = 2, 3, · · · ,
(3.20)
n = 1, 2, · · · ,
(3.21)
k=0
respectively. The above examples illustrate that the operational relations established in this paper can be used to derive the results for the newly introduced families of polynomials from the results of any member belonging to the Appell family.
S. Khan, M. Riyasat / J. Math. Anal. Appl. 421 (2015) 806–829
822
Table 2 Some known Sheffer polynomials. S. No.
A(t); H(t) −t2
g(t); f (t)
I.
e
; 2t
e
II.
e−t ; νt
III.
(1 − t)−1 ;
IV.
(1 − t)−α−1 ;
t2 4
;
t
m
m
e( ν ) ; t t−1
t t−1
1+t ln( 1−t )
Generating functions 2 tn e2xt−t = ∞ n=0 Hn (x) n!
t 2
exp(νxt − tm ) =
t ν
(1 − t)−1 ;
t t−1
exp(βt + x(1 − et )) =
exp(a(et −1)); a(et − 1)
e−t (1 +
λ −μ x (1 + (1 + t) ) (1 + t)tn = ∞ s (x; λ, μ) n n=0 n! ∞ x t tn n=0 bn (x) n! ln(1+t) (1 + t) =
e−t ; ln(1 +
VIII.
(1 + (1 + t)λ )−μ ; ln(1 + t)
(1 + eλt )μ ; et − 1
IX.
t ln(1+t) ;
t et −1 ;
X.
2 2+t ;
XI.
1 1+t2
XII.
(1 − 4t) 2 × ( 1+√21−4t )a−1 ; (1+
−4t √ 1−4t)2
∞
et − 1
xt exp ( t−1 )=
=
t x a)
∞
=
n=0
n=0
Laguerre polynomials n!Ln (x) [1]
n L(α) n (x)t
Generalized Laguerre polynomials n!L(α) n (x) [1,25]
n
Pn (x) tn!
∞
n
n=0
t a(β) n (x) n!
Actuarial polynomials a(β) n (x) [6]
cn (x; a) tn!
+ et ); e −1
2 2+t (1
+ t)x =
sec t; tan t
1 1+t2
exp(x arctan(t)) =
n=0
Pidduck polynomials Pn (x) [6,16]
n
n=0
∞
∞
1 2 (1 t
1+t (1−t)a ; 1 1 1+t 2 4 − 4 ( 1−t )
Generalized Hermite polynomials Hn,m,ν (x) [20]
Ln (x)tn
(1 − t)−β ; ln(1 − t)
VII.
−1
n=0
1+t x t 1−t ( 1−t )
eβt ; 1 − et
; arctan(t)
∞
et −1 2 et −1 ; et +1
VI.
ln(1 + t)
Hn,m,ν (x) tn!
1 (1−t)α+1
t t−1
t 1−t ;
ln(1 + t)
n
n=0
xt exp ( t−1 )=
Hermite polynomials Hn (x) [1]
(1 − t)−α−1 ;
V.
t a)
1 (1−t)
∞
Polynomials
Poisson–Charlier polynomials cn (x; a) [15,17,29] Peters polynomials sn (x; λ, μ) [6] Bernoulli polynomials of the second kind bn (x) [17]
n
rn (x) tn! ∞ n=0
−1
Related polynomials rn (x) [17] n
Rn (x) tn!
√ (1 − 4t) 2 ( 1+√21−4t )a−1 exp( (1+−4xt ) 1−4t)2 n R (a, x)t = ∞ n n=0
Hahn polynomials Rn (x) [4] Shively’s pseudo-Laguerre polynomials Rn (a, x) [25]
Acknowledgments The authors are thankful to the reviewer(s) for several useful comments and suggestions towards the improvement of this paper. The authors would also like to express their thanks to the editor for encouraging comments. Appendix A The Appell and Sheffer polynomial sequences arise in numerous problems of applied mathematics, theoretical physics, approximation theory and several other mathematical branches. A list of polynomials belonging to the Appell family is given in Table 1. We present the list of some known Sheffer polynomials in Table 2. The Hermite and Laguerre polynomials are important members of the Sheffer family. Taking A(t) (or 1 −t2 and H(t) (or f −1 (t)) = 2t of the Hermite polynomials Hn (x) (Table 2(I)) in Eq. (2.1a) (or g(f −1 (t)) ) = e (2.1b)), we get the following generating function for the Hermite–Appell polynomials [19]: ∞
tn , n!
(A.1a)
∞ tn 1 (2xt−t2 ) e . = H An (x) g(t) n! n=0
(A.1b)
2
A(t)e(2xt−t
)
=
n=0
H An (x)
or, equivalently
S. Khan, M. Riyasat / J. Math. Anal. Appl. 421 (2015) 806–829
Table 3 Results for
H An (x)
and
823
L An (x).
S. No.
Results
I.
Series definitions
H An (x)
Hermite–Appell polynomials H An (x)
n = n k=0 k Ak Hn−k (x)
II.
Multiplicative and derivative operators
ˆ HA = 2x − Dx + M PˆHA =
Laguerre–Appell polynomials L An (x) n n L An (x) = k=0 k Ak Ln−k (x)
A (Dx /2) A(Dx /2)
ˆ LA = −xD 2 + (2x − 1)Dx − (x − 1) M x (Dx /Dx −1) + A A(Dx /Dx −1) PˆLA = Dx
Dx 2
Dx −1
III.
Differential equations
(Dx /2) 2 (Dx − 2xDx − A A(Dx /2) Dx + 2n)H An (x) = 0
2 (xDx − (2x − 1)Dx + (x − 1) (Dx /Dx −1) Dx −1 − A A(Dx /Dx −1) + n Dx )L An (x) = 0
IV.
Operational rules
ˆ H An (x) = An (MH ); ˆ H := Multiplicative operator of Hn (x) M
ˆ L An (x) = An (ML ); ˆ L := Multiplicative operator of Ln (x) M
1 1 t −1 Similarly, taking A(t) (or g(f −1 (t)) = t−1 of the Laguerre polynomials Ln (x) (t)) ) = 1−t and H(t) (or f (Table 2(III)) in Eq. (2.1a) (or (2.1b)), we get the following generating function for the Laguerre–Appell polynomials [18]: ∞ 1 ( −xt tn e 1−t ) = , L An (x) 1−t n! n=0
(A.2a)
∞ 1 ( −xt tn 1 e 1−t ) = . L An (x) g(t) 1 − t n! n=0
(A.2b)
A(t) or, equivalently
Certain results for H An (x) and L An (x), which are not considered in [19,18] are given in Table 3. Taking sn (x) = Hn (x) (n = 0, 1, · · ·) (Table 2(I)) in Eqs. (2.7) and (2.8), we get the following determinantal definition of the Hermite–Appell polynomials H An (x): Definition A.1. The Hermite–Appell polynomials H A0 (x)
=
1 , β0
β0 =
n
H An (x)
βn = −
=
1 A0
(−1)
n+1
(β0 )
1 β 0 0 0 . . 0
H An (x)
of degree n are defined by
1 , A0 H1 (x) β1
(A.3) H2 (x) · · ·
Hn−1 (x)
···
β0
β2
2 1 β1
0
β0
.
.
···
.
.
.
···
.
0
···
β0
0
n n Ak βn−k , k
βn−1
n−1 ··· 1 βn−2
n−1 ··· 2 βn−3
Hn (x) βn
n βn−1
n1 , 2 βn−2 . .
n n−1 β1
n = 1, 2, 3, · · · ,
(A.4)
k=1
where β0 , β1 , · · · , βn ∈ R, β0 = 0 and Hn (x) (n = 0, 1, · · ·) are the Hermite polynomials of degree n. Similarly, taking sn (x) = Ln (x) (n = 0, 1, · · ·) (Table 2(III)) in Eqs. (2.7) and (2.8), we get the following determinantal definition of the Laguerre–Appell polynomials L An (x):
S. Khan, M. Riyasat / J. Math. Anal. Appl. 421 (2015) 806–829
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Table 4 Results for
H Bn (x)
and
H En (x).
S. No.
Results
I.
Series definitions
H Bn (x)
II.
Multiplicative and derivative operators
ˆ HB = 2x − Dx + M
Differential equations
2 (Dx − 2xDx −
Operational rules
+ 2n)H Bn (x) = 0 ˆ H Bn (x) = Bn (MH ); ˆ H := Multiplicative operator of Hn (x) M
III. IV.
Hermite–Bernoulli polynomials H Bn (x)
n = n k=0 k Bk Hn−k (x)
PˆHB =
eDx /2 (1− Dx )−1
ˆ HE = 2x − Dx − M
2
Dx 2
Dx 2
(eDx /2 −1)
PˆHE =
eDx /2 (1− Dx )−1 2
Dx 2
Hermite–Euler polynomials H En (x) n n H En (x) = k=0 k Ek Hn−k (x)
(eDx /2 −1)
2 (Dx − 2xDx +
Dx
eDx /2 eDx /2 +1
Dx 2 eDx /2 eDx /2 +1
+ 2n)H En (x) = 0
ˆ H En (x) = En (MH ); ˆ H := Multiplicative operator of Hn (x) M
Definition A.2. The Laguerre–Appell polynomials L An (x) of degree n are defined by L A0 (x)
=
1 , β0
β0 =
n
L An (x)
βn = −
=
1 A0
(−1)
n+1
(β0 )
n k=1
1 β 0 0 0 . . 0
1 , A0
(A.5)
L1 (x) L2 (x) · · · β1
Ln−1 (x)
···
β0
β2
2 1 β1
0
β0
.
.
···
.
.
.
···
.
0
···
β0
0
n Ak βn−k , k
βn−1
n−1 ··· 1 βn−2
n−1 ··· 2 βn−3
Ln (x) βn
n 1 βn−1
n , 2 βn−2 . .
n n−1 β1
n = 1, 2, 3, · · · ,
(A.6)
where β0 , β1 , · · · , βn ∈ R, β0 = 0 and Ln (x) (n = 0, 1, · · ·) are the Laguerre polynomials of degree n. With the help of the results given in Table 3 and by taking A(t) (or g(t)) of the members belonging to the Appell polynomials family, we can derive the generating functions and other results for the corresponding members of the Hermite–Appell and Laguerre–Appell families. 1 t To give an example, we take An (x) = Bn (x) then from Table 1(I), we have A(t) (or g(t) ) = et −1 and thus in view of Eq. (A.1a) (or (A.1b)), we get the following generating function for the Hermite–Bernoulli polynomials H Bn (x): ∞ tn t (2xt−t2 ) e . = H Bn (x) t e −1 n! n=0
(A.7)
1 Similarly, taking An (x) = En (x) (Table 1(II)), we have A(t) (or g(t) ) = et2+1 and thus in view of Eq. (A.1a) (or (A.1b)), we get the following generating function for the Hermite–Euler polynomials H En (x): ∞ 2 tn (2xt−t2 ) e . = H En (x) t e +1 n! n=0
(A.8)
The series definitions and other results for the Hermite–Bernoulli and Hermite–Euler polynomials are given in Table 4. 1 Taking β0 = 1 and βi = i+1 (i = 1, 2, · · · , n) in Eqs. (A.3) and (A.4), we find the following determinantal definition of the Hermite–Bernoulli polynomials H Bn (x):
S. Khan, M. Riyasat / J. Math. Anal. Appl. 421 (2015) 806–829
Definition A.3. The Hermite–Bernoulli polynomials H B0 (x)
H Bn (x)
of degree n are defined by
= 1,
1 1 0 n H Bn (x) = (−1) 0 . . 0
H1 (x) H2 (x) · · ·
Hn−1 (x)
···
1
1
23 1 1 2
0
1
···
.
.
···
.
.
.
···
.
0
0
···
1
1 2
···
825
1
n−1n
1
1
n−1 n−1 1 2
n−2
Hn (x) 1
n+1 n 1 1 n
n 1 , 2 n−1 . .
n 1
(A.9)
n = 1, 2, · · · ,
(A.10)
n−1 2
where Hn (x) (n = 0, 1, 2, · · ·) are the Hermite polynomials of degree n. Next, taking β0 = 1 and βi = 12 (i = 1, 2, · · · , n) in Eqs. (A.3) and (A.4), we find the following determinantal definition of the Hermite–Euler polynomials H En (x): Definition A.4. The Hermite–Euler polynomials H E0 (x)
H En (x)
of degree n are defined by
= 1,
1 1 0 n E (x) = (−1) 0 H n . . 0
H1 (x) H2 (x) · · · Hn−1 (x) 1 2
1
···
1
1
2 1 2
2 1 n−1
2 1
···
0
1
···
2
1 1 n−1 2 2
.
.
···
.
.
.
···
.
0
0
···
1
Hn (x) 1 2
1 n 2 1
1 n , 2 2 . .
n 1
(A.11)
n = 1, 2, · · · ,
(A.12)
2 n−1
where Hn (x) (n = 0, 1, 2, · · ·) are the Hermite polynomials of degree n. Also, in view of Eq. (A.2a) (or (A.2b)), we get the following generating functions for the Laguerre– Bernoulli and Laguerre–Euler polynomials L Bn (x) and L En (x): ∞ tn t ( −xt 1−t ) = e L Bn (x) t (e − 1)(1 − t) n! n=0
(A.13)
∞ tn 2 ( −xt 1−t ) = e , L En (x) t (e + 1)(1 − t) n! n=0
(A.14)
and
respectively. The series definitions and other results for the Laguerre–Bernoulli and Laguerre–Euler polynomials are given in Table 5. 1 Taking β0 = 1 and βi = i+1 (i = 1, 2, · · · , n) in Eqs. (A.5) and (A.6), we find the following determinantal definition of the Laguerre–Bernoulli polynomials L Bn (x):
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Table 5 Results for
L Bn (x)
and
L En (x).
S. No.
Results
I.
Series definitions
L Bn (x)
Laguerre–Bernoulli polynomials
n = n k=0 k Bk Ln−k (x)
II.
Multiplicative and derivative operators
ˆ LB = −xD 2 + (2x − 1)Dx − (x − 1) M x +
exp ( (
PˆLB = III.
Differential equations
Dx Dx −1
Dx Dx −1
L Bn (x)
Laguerre–Euler polynomials L En (x) n n L En (x) = k=0 k Ek Ln−k (x) ˆ LE = −xD 2 + (2x − 1)Dx − (x − 1) M x
Dx )−1 Dx −1 Dx )−1) Dx −1
)(1−
)(exp (
−
PˆLE =
Dx Dx −1
2 (xDx − (2x − 1)Dx + (x − 1)
−
+
Operational rules
ˆ L Bn (x) = Bn (ML ); ˆ L := Multiplicative operator of Ln (x) M
Dx ) Dx −1 Dx )+1) Dx −1
exp ( (exp (
x −1 + n DD )L Bn (x) = 0 x
IV.
Dx Dx −1
2 (xDx − (2x − 1)Dx + (x − 1)
Dx )(1− Dx )−1 Dx −1 Dx −1 Dx )(exp( Dx )−1) Dx −1 Dx −1
exp ( (
Dx ) Dx −1 Dx )+1) Dx −1
exp ( (exp (
x −1 + n DD )L En (x) = 0 x
ˆ L En (x) = En (ML ); ˆ L := Multiplicative operator of Ln (x) M
Definition A.5. The Laguerre–Bernoulli polynomials L Bn (x) of degree n are defined by L B0 (x)
= 1,
1 1 0 n B (x) = (−1) 0 L n . . 0
L1 (x) L2 (x)
···
Ln−1 (x)
···
1
1
23 1 1 2
0
1
···
.
.
···
.
.
.
···
.
0
0
···
1
1 2
···
1
n−1n
1
1
n−1 n−1 1 2
n−2
Ln (x) 1
n+1 n 1 1 n
n 1 , 2 n−1 . .
n 1
(A.15)
n = 1, 2, · · · ,
(A.16)
n−1 2
where Ln (x) (n = 0, 1, 2, · · ·) are the Laguerre polynomials of degree n. Taking β0 = 1 and βi = 12 (i = 1, 2, · · ·) in Eqs. (A.5) and (A.6), we find the following determinantal definition of the Laguerre–Euler polynomials L En (x): Definition A.6. The Laguerre–Euler polynomials L En (x) of degree n are defined by L E0 (x)
= 1,
1 1 0 n L En (x) = (−1) 0 . . 0
L1 (x) L2 (x)
· · · Ln−1 (x) ···
1
1 2
1 2 2 1
0
1
···
2
1 1 n−1 2 2
.
.
···
.
.
.
···
.
0
0
···
1
1 2
···
1
2 1 n−1
Ln (x) 1 2
1 n 2 1
n 1 , 2 2 . .
n 1
(A.17)
n = 1, 2, · · · ,
(A.18)
2 n−1
where Ln (x) (n = 0, 1, 2, · · ·) are the Laguerre polynomials of degree n. Further, we proceed to draw the graphs of H Bn (x), H En (x), L Bn (x) and L En (x). To draw the graphs of these polynomials, we find the values of the first five Hermite polynomials Hn (x) and Laguerre polynomials Ln (x) by making use of their generating functions given in Table 2(I) and (III). We give the list of first five Hermite and Laguerre polynomials in Table 6.
S. Khan, M. Riyasat / J. Math. Anal. Appl. 421 (2015) 806–829
827
Table 6 First five Hn (x) and Ln (x). n
0
1
2
3
4
Hn (x) Ln (x)
1 1
2x 1−x
4x2 − 2 2 1 2 (x − 4x + 2)
8x3 − 12x 3 2 1 6 (−x + 9x − 18x + 6)
16x4 − 48x2 + 12 4 3 2 1 24 (x − 16x + 72x − 96x + 24)
Next, we consider the values of H Bn (x), (A.10), (A.12), (A.16) and (A.18), we have
H En (x), L Bn (x)
and L En (x) for n = 4. Taking n = 4 in Eqs.
1 H1 (x) H2 (x) H3 (x) H4 (x) 1/2 1/3 1/4 1/5 1 1 1 1 1 , H B4 (x) = 0 0 1 3/2 2 0 0 0 0 1 2 1 H1 (x) H2 (x) H3 (x) H4 (x) 1/2 1/2 1/2 1/2 1 1 1 3/2 2 , H E4 (x) = 0 0 1 3/2 3 0 0 0 0 1 2 1 L1 (x) L2 (x) L3 (x) L4 (x) 1/3 1/4 1/5 1 1/2 1 1 1 1 L B4 (x) = 0 0 1 3/2 2 0 0 0 0 1 2
(A.19)
(A.20)
(A.21)
and 1 L1 (x) L2 (x) L3 (x) L4 (x) 1/2 1/2 1/2 1 1/2 1 1 3/2 2 , L E4 (x) = 0 0 1 3/2 3 0 0 0 0 1 2
(A.22)
respectively. Now, by using the expressions of first five Hn (x) and Ln (x) from Table 6 in Eqs. (A.19)–(A.22) and simplifying, we find
H B4 (x)
= 16x4 − 16x3 − 44x2 + 24x +
H E4 (x)
299 , 30
= 16x4 − 16x3 − 48x2 + 26x + 12, 1 4 1 3 1 2 1 x − x + x − , 24 3 2 30 1 4 1 3 x − x + x. L E4 (x) = 24 3
L B4 (x)
=
(A.23) (A.24) (A.25) (A.26)
828
S. Khan, M. Riyasat / J. Math. Anal. Appl. 421 (2015) 806–829
Using Eqs. (A.23)–(A.26), we get the following graphs:
We know that the Bernoulli numbers Bn [25] appear in Taylor series expansions of the tangent and hyperbolic tangent functions, while the Euler numbers En [25] appear in Taylor series expansions of the secant and hyperbolic secant functions. The Bernoulli and Euler numbers have deep connections with number theory and occur in combinatorics. These numbers also appear as special values of the Bernoulli and Euler polynomials. We note that Bn = Bn (0) and 1 En = 2 En 2 n
The importance of the Bernoulli and Euler numbers provides motivation to introduce the numbers related to the polynomial families introduced in this article. This aspect will be taken in next investigation. References [1] L.C. Andrews, Special Functions for Engineers and Applied Mathematicians, Macmillan Publishing Company, New York, 1985.
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[2] T.M. Apostol, On the Lerch zeta function, Pacific J. Math. 1 (1951) 161–167. [3] P. Appell, Sur une classe de polynômes, Ann. Sci. Ec. Norm. Super. 9 (2) (1880) 119–144. [4] C.M. Bender, Solution of operator equations of motion, in: J. Dittrich, P. Exner (Eds.), Rigorous Results in Quantum Dynamics, World Scientific, Singapore, 1991, pp. 99–112. [5] P. Blasiack, G. Dattoli, H. Horzela, K. Penson, Representations of monomiality principle with Sheffer type polynomials and boson normal ordering, Phys. Lett. A 352 (2006) 7–12. [6] R.P. Boas, R.C. Buck, Polynomial Expansions of Analytic Functions, Springer-Verlag, Berlin, Gottingen, Heidelberg, 1958. [7] G. Bretti, P. Natalini, P.E. Ricci, Generalizations of the Bernoulli and Appell polynomials, Abstr. Appl. Anal. 7 (2004) 613–623. [8] F.A. Costabile, F. Dell’Accio, M.I. Gualtieri, A new approach to Bernoulli polynomials, Rend. Mat. Appl. 26 (1) (2006) 1–12. [9] F.A. Costabile, E. Longo, A determinantal approach to Appell polynomials, J. Comput. Appl. Math. 234 (5) (2010) 1528–1542. [10] F.A. Costabile, E. Longo, An algebraic approach to Sheffer polynomial sequences, Integral Transforms Spec. Funct. 25 (4) (2013) 295–311. [11] G. Dattoli, Hermite–Bessel and Laguerre–Bessel functions: a by-product of the monomiality principle, in: Advanced Special Functions and Applications, Melfi, 1999, in: Proc. Melfi Sch. Adv. Top. Math. Phys., vol. 1, Aracne, Rome, 2000, pp. 147–164. [12] G. Dattoli, S. Lorenzutta, D. Sacchetti, Integral representations of new families of polynomials, Ital. J. Pure Appl. Math. 15 (19–28) (2004) 19–28. [13] G. Dattoli, M. Migliorati, H.M. Srivastava, Sheffer polynomials, monomiality principle, algebraic methods and the theory of classical polynomials, Math. Comput. Modelling 45 (9–10) (2007) 1033–1041. [14] K. Douak, The relation of the d-orthogonal polynomials to the Appell polynomials, J. Comput. Appl. Math. 70 (2) (1996) 279–295. [15] A. Erdélyi, W. Magnus, F. Oberhettinger, F.G. Tricomi, Higher Transcendental Functions, Vol. II, McGraw–Hill Book Company, New York, Toronto and London, 1953. [16] A. Erdélyi, W. Magnus, F. Oberhettinger, F.G. Tricomi, Higher Transcendental Functions, Vol. III, McGraw–Hill Book Company, New York, Toronto and London, 1955. [17] C. Jordan, Calculus of Finite Differences, third edition, Chelsea Publishing Company, Bronx, New York, 1965. [18] S. Khan, M.W.M. Al-Saad, R. Khan, Laguerre-based Appell polynomials: properties and applications, Math. Comput. Modelling 52 (1–2) (2010) 247–259. [19] S. Khan, G. Yasmin, R. Khan, N.A.M. Hassan, Hermite-based Appell polynomials: properties and applications, J. Math. Anal. Appl. 351 (2) (2009) 756–764. [20] M. Lahiri, On a generalisation of Hermite polynomials, Proc. Amer. Math. Soc. 27 (1971) 117–121. [21] Q.M. Luo, Apostol–Euler polynomials of higher order and the Gaussian hypergeometric function, Taiwanese J. Math. 10 (4) (2006) 917–925. [22] Q.M. Luo, H.M. Srivastava, Some generalizations of the Apostol–Bernoulli and Apostol–Euler polynomials, J. Math. Anal. Appl. 308 (1) (2005) 290–302. [23] W. Magnus, F. Oberhettinger, R.P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics, Springer-Verlag, New York, 1966. [24] K.A. Penson, P. Blasiak, G. Dattoli, G.H.E. Duchamp, A. Horzela, A.I. Solomon, Monomiality principle, Sheffer type polynomials and the normal ordering problem, in: The Eighth International School on Theoretical Physics “Symmetry and Structural Properties of Condensed Matter”, Myczkowce, Poland, August 2005, in: J. Phys. Conf. Ser., vol. 30, 2006, pp. 86–97. [25] E.D. Rainville, Special Functions, reprint of the 1960 first edition, Chelsea Publishing Company, Bronx, New York, 1971. [26] S. Roman, The Umbral Calculus, Academic Press, New York, 1984. [27] I.M. Sheffer, Some properties of polynomial sets of type zero, Duke Math. J. 5 (1939) 590–622. [28] J.F. Steffensen, The poweriod, an extension of the mathematical notion of power, Acta Math. 73 (1941) 333–366. [29] G. Szegö, Orthogonal Polynomials, Amer. Math. Soc., Providence, RI, 1978.
Contents lists available at ScienceDirect
Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa
A determinantal approach to Sheffer–Appell polynomials via monomiality principle Subuhi Khan ∗ , Mumtaz Riyasat Department of Mathematics, Aligarh Muslim University, Aligarh, India
a r t i c l e
i n f o
Article history: Received 20 April 2014 Available online 28 July 2014 Submitted by M.J. Schlosser Keywords: Sheffer–Appell polynomials Determinantal approach Monomiality principle Sheffer–Bernoulli polynomials Sheffer–Euler polynomials
a b s t r a c t In this article, the Sheffer and Appell polynomials are combined to introduce the family of Sheffer–Appell polynomials by using operational methods. The determinantal definition and other properties of the Sheffer–Appell polynomials are established. As particular cases of these polynomials, the Sheffer–Bernoulli and Sheffer–Euler polynomials are introduced and their determinantal definitions are obtained. The operational correspondence between the Appell and Sheffer–Appell polynomials is used to derive the results for the Sheffer–Appell polynomials. Certain results for the Hermite–Appell and Laguerre–Appell polynomials are also obtained. © 2014 Elsevier Inc. All rights reserved.
1. Introduction and preliminaries One of the important classes of polynomial sequences is the class of Sheffer sequences [27]. The class of Appell polynomial sequences [3], which is a subclass of Sheffer polynomial sequences is equally important. The Appell and Sheffer polynomial sequences arise in numerous problems of applied mathematics, theoretical physics, approximation theory and several other mathematical branches. In the past few decades, there has been a renewed interest in Appell and Sheffer sequences. Properties of the Appell and Sheffer sequences are naturally handled within the framework of modern classical umbral calculus by Roman [26]. We recall that, in 1880, Appell [3] introduced and studied sequences of n-degree polynomials An (x),
n = 0, 1, 2, · · · ,
(1.1)
satisfying the recurrence relation d An (x) = nAn−1 (x), dx
n = 1, 2, · · · .
* Corresponding author. E-mail addresses: [email protected] (S. Khan), [email protected] (M. Riyasat). http://dx.doi.org/10.1016/j.jmaa.2014.07.044 0022-247X/© 2014 Elsevier Inc. All rights reserved.
(1.2)
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807
In particular, Appell noticed the one-to-one correspondence of the set of such sequences {An (x)}n and the set of numerical sequences {An }n , A0 = 0 given by the explicit representation An (x) = An +
n n An−1 x + An−2 x2 + · · · + A0 xn , 1 2
n = 0, 1, 2, · · · .
(1.3)
The above equation, in particular shows explicitly that for each n ≥ 1, An (x) is completely determined by An−1 (x) and by the choice of the constant of integration An . Appell also provided an alternate general method to determine such sequences of polynomials that satisfy relation (1.2). In fact, given the power series
A(t) = A0 +
∞ t t2 tn tn A 1 + A2 + · · · + A n + · · · = An , 1! 2! n! n! n=0
A0 = 0,
(1.4)
with Ai (i = 0, 1, 2, · · ·) real coefficients, a sequence of polynomials satisfying (1.2) is determined by the power series expansion of the product A(t)ext , that is, A(t)ext = A0 (x) +
t t2 tn A1 (x) + A2 (x) + · · · + An (x) + · · · , 1! 2! n!
or, equivalently A(t)ext =
∞
An (x)
n=0
tn . n!
(1.5)
The function A(t)ext is called the generating function of the sequence of polynomials An (x). Alternatively, the sequence An (x) is Appell for g(t) [26], if and only if ∞ tn 1 xt e = An (x) , g(t) n! n=0
(1.6)
where g(t) =
∞
gn
n=0
tn , n!
g0 = 0.
(1.7)
In view of relations (1.5) and (1.6), we have A(t) =
1 . g(t)
(1.8)
There are several ways to define the Sheffer sequences [27], among which by a generating function and by a differential recurrence relation are most common. A polynomial sequence {sn(x)}∞ n=0 (sn (x) being a polynomial of degree n) is called Sheffer A-type zero [25, p. 222 (Theorem 72)] (which we shall hereafter call Sheffer-type), if sn (x) possesses the exponential generating function of the form A(t)exH(t) =
∞
sn (x)
n=0
where A(t) and H(t) have (at least the formal) expansions:
tn , n!
(1.9)
808
S. Khan, M. Riyasat / J. Math. Anal. Appl. 421 (2015) 806–829 ∞
A(t) =
An
tn , n!
A0 = 0
(1.10a)
Hn
tn , n!
H1 = 0,
(1.10b)
n=0
and ∞
H(t) =
n=1
respectively. We know that the order of a formal power series f (t) =
∞
ak t k ,
ak ∈ C,
k=0
denoted by O(f (t)) is the smallest integer k for which the coefficient of tk does not vanish. If O(f (t)) = 1, then the formal power series f (t) has a compositional inverse f¯(t) satisfying f (f¯(t)) = f¯(f (t)) = t. A series f (t) for which O(f (t)) = 1 is called a delta series. We recall the following result [26, p. 17], which can be viewed as an alternate definition of Sheffer sequences. Let f (t) be a delta series and g(t) be an invertible series of the following forms: f (t) =
∞
fn
n=0
tn , n!
f0 = 0, f1 = 0
(1.11a)
and g(t) =
∞
gn
n=0
tn , n!
g0 = 0.
(1.11b)
Then there exists a unique sequence sn (x) of polynomials satisfying the orthogonality conditions g(t)f (t)k sn (x) = n!δn,k ,
∀n, k ≥ 0.
(1.12)
According to Roman [26, p. 18 (Theorem 2.3.4)], the polynomial sequence sn (x) is uniquely determined by two (formal) power series given by Eqs. (1.11a) and (1.11b). The exponential generating function of sn (x) is then given by 1 g(f −1 (t))
ex(f
−1
(t))
=
∞
sn (x)
n=0
tn , n!
(1.13)
for all x in C, where f −1 (t) is the compositional inverse of f (t). In view of Eqs. (1.9) and (1.13), we have A(t) =
1 g(f −1 (t))
(1.14)
and H(t) = f −1 (t).
(1.15)
The sequence sn (x) in Eq. (1.12) is the Sheffer sequence for the pair (g(t), f (t)). The Sheffer sequence for (1, f (t)) is called the associated Sheffer sequence for f (t) and the Sheffer sequence for (g(t), t) becomes
S. Khan, M. Riyasat / J. Math. Anal. Appl. 421 (2015) 806–829
809
the Appell sequence for g(t) [26, p. 17]. Also, from Eqs. (1.9) (or (1.13)) and (1.5) (or (1.6)), it follows that for H(t) = f −1 (t), the Sheffer polynomials sn (x) reduce to the Appell polynomials An (x). ˆ and Pˆ Next, we recall that according to the monomiality principle [28,11], there exist two operators M playing, respectively, the role of multiplicative and derivative operators for a polynomial set {pn (x)}n∈N , ˆ and Pˆ satisfy the following identities, for all n ∈ N: that is, M ˆ pn (x) = pn+1 (x) M
(1.16)
Pˆ pn (x) = npn−1 (x).
(1.17)
and
ˆ and Pˆ must satisfy the The polynomial set {pn (x)} is then called a quasi-monomial. The operators M commutation relation ˆ ] = Pˆ M ˆ −M ˆ Pˆ = ˆ1 [Pˆ , M
(1.18)
and thus display a Weyl group structure. If the considered polynomial set {pn (x)} is quasi-monomial, its properties can be easily derived from ˆ and Pˆ operators. In fact the following hold: each of the M ˆ and Pˆ have differential realizations, then the polynomials pn (x) satisfy the differential equation (i) If M ˆ Pˆ pn (x) = npn (x). M
(1.19)
(ii) Assuming that p0 (x) = 1, then pn (x) can be explicitly constructed as ˆ n {1}. pn (x) = M
(1.20)
(iii) In view of identity (1.20) the exponential generating function of pn (x) can be given in the form ˆ
etM {1} =
∞
pn (x)
n=0
tn , n!
|t| < ∞.
(1.21)
The Appell polynomials An (x) are quasi-monomial with respect to the following multiplicative and derivative operators: ˆ A = x + A (Dx ) , M A(Dx )
(1.22a)
ˆ A = x − g (Dx ) M g(Dx )
(1.22b)
or, equivalently
and PˆA = Dx , respectively.
(1.23)
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The Sheffer polynomials sn (x) and the monomiality principle, along with the underlying operational formalism, offer a powerful tool for investigation of the properties of a wide class of polynomials, see for example [5,13,24]. It has been shown in [24], that if sn (x) are of Sheffer-type then it is possible to give ˆ and Pˆ . Conversely, if M ˆ =M ˆ (x, Dx ) and Pˆ = Pˆ (Dx ), then sn (x) satisfying explicit representations of M Eqs. (1.16) and (1.17) are necessarily of Sheffer-type. The multiplicative and derivative operators (also known as raising and lowering operators) for the Sheffer polynomials sn (x) are given by −1
ˆ s = xH H −1 (Dx ) + A (H (Dx )) , M A(H −1 (Dx ))
(1.24a)
or, equivalently g (Dx ) 1 x− g(Dx ) f (Dx )
ˆs = M
(1.24b)
and Pˆs = H −1 (Dx ),
(1.25a)
Pˆs = f (Dx ),
(1.25b)
or, equivalently
respectively. Costabile et al. [8] has given a new approach to Bernoulli polynomials based on a determinantal definition. This approach is further extended to provide determinantal definitions of the Appell and Sheffer polynomial sequences by Costabile and Longo in [9] and [10] respectively. The equivalence of determinantal approach with other existing approaches is also established. The simplicity of the algebraic approach to the Appell and Sheffer sequences established in [9,10], allows several applications. The above mentioned research works of Costabile and Longo and the importance of operational methods in the theory of special functions motivated the authors to introduce and study the Sheffer–Appell polynomials by using operational techniques and determinantal approach. In this paper, the Sheffer–Appell polynomials are introduced by means of generating function, series definition and determinantal definition. These polynomials are framed within the context of monomiality principle and their properties are derived. The corresponding results for the Sheffer–Bernoulli and Sheffer– Euler polynomials are also obtained. Examples of some members belonging to the family of Sheffer–Appell polynomials are considered and the graphs of certain members are drawn for suitable values of the indices. 2. Sheffer–Appell polynomials The polynomials defined as the discrete convolution of the known polynomials are used to explore new (A) families of special polynomials. The discrete Appell convolution fn (x) is defined as fn(A) (x)
=
n n k=0
k
Ak fn−k (x). (A)
Taking the generic polynomials fn (x) (n ∈ N, x ∈ R) as xn in the above equation, we find that fn (x) reduce to the Appell polynomials An (x) [26]. The Sheffer–Appell polynomials denoted by s An (x) are defined as the discrete Appell convolution of the Sheffer polynomials sn (x).
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In this section, we introduce the Sheffer–Appell polynomials by means of generating function and series definition. Further, a determinantal definition of the Sheffer–Appell polynomials is given. First, we derive the generating function for the Sheffer–Appell polynomials by proving the following result. Theorem 2.1. The generating function for the Sheffer–Appell polynomials s An (x) is given as: A(t)A(t)exH(t) =
∞
s An (x)
n=0
tn , n!
(2.1a)
or, equivalently ∞ 1 x(f −1 (t)) tn e . = s An (x) −1 g(f (t)) g(t) n! n=0
1
(2.1b)
ˆ s of the Sheffer polynomials sn (x) in the l.h.s. of Proof. Replacing x by the multiplicative operator M generating function (1.5) of the Appell polynomials and denoting the resultant Sheffer–Appell polynomials in the r.h.s. by s An (x), we have ˆ
A(t)eMs t {1} =
∞ n=0
s An (x)
tn , n!
(2.2)
which in view of Eq. (1.21) gives A(t)
∞
sn (x)
n=0
∞ tn tn = . s An (x) n! n=0 n!
(2.3)
Using generating function (1.9) of the Sheffer polynomials sn (x) in the l.h.s. of the above equation we get assertion (2.1a). Also, in view of relations (1.8), (1.14) and (1.15), Eq. (2.1a) can be expressed equivalently as (2.1b). 2 Next, we obtain the series definition of the Sheffer–Appell polynomials s An (x) by proving following result. Theorem 2.2. The Sheffer–Appell polynomials s An (x) are defined by the series: s An (x) =
n n Ak sn−k (x). k
(2.4)
k=0
Proof. Using expansion (1.4) of A(t) in the l.h.s. of Eq. (2.3), we find ∞ ∞ n=0 k=0
Ak sn (x)
∞ tn+k tn = . s An (x) n!k! n! n=0
(2.5)
Replacing n by n − k in the l.h.s. of Eq. (2.5) and then equating the coefficients of like powers of t in both sides of the resultant equation, we get assertion (2.4). 2 Note. It is important to note that definition (2.4) is a direct consequence of the fact that Sheffer–Appell polynomials s An (x) are discrete Appell convolution of the Sheffer polynomials sn (x).
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812
Remark 2.1. From Eqs. (1.5) and (2.2), we get the following operational correspondence between the Sheffer– Appell polynomials s An (x) and Appell polynomials An (x): ˆ s ). = An (M
s An (x)
(2.6)
In order to give the determinantal definition of the Sheffer–Appell polynomials s An (x), we prove the following result. Theorem 2.3. The Sheffer–Appell polynomials s An (x) of degree n are defined by
s A0 (x)
=
1 , β0
β0 =
n
s An (x)
=
(−1)
n+1
(β0 )
1 β 0 0 0 . . 0
1 , A0
(2.7)
s1 (x)
s2 (x)
···
sn−1 (x)
β1
···
β0
β2
2 1 β1
0
β0
···
βn−1
n−1 1 βn−2
n−1 2 βn−3
.
.
···
.
.
.
···
.
0
0
···
β0
···
sn (x) βn
n 1 βn−1
n , 2 βn−2 . .
n n−1 β1
1 βn = − A0
n n k=1
n = 1, 2, 3, · · · ,
k
Ak βn−k ,
(2.8)
where β0 , β1 , · · · , βn ∈ R, β0 = 0 and sn (x) (n = 0, 1, · · ·), are the Sheffer polynomials defined by Eq. (1.9) (or (1.13)). Proof. First, we consider the following determinantal definition of the Appell polynomials An (x) of degree n given by Costabile and Longo [9, p. 1533]:
A0 (x) =
1 , β0
β0 =
n
An (x) =
(−1)
n+1
(β0 )
n = 1, 2, 3, · · · ,
1 β 0 0 0 . . 0
1 , A0
(2.9)
x
x2
···
β1
···
β0
β2
2 1 β1
0
β0
.
.
···
.
.
.
···
.
0
0
···
β0
xn−1
βn−1
n−1 ··· 1 βn−2
n−1 ··· 2 βn−3
βn
n βn−1
n1 , 2 βn−2 . .
n β xn
n−1
n
1 n βn = − Ak βn−k , A0 k k=1
1
(2.10)
where β0 , β1 , · · · , βn ∈ R, β0 = 0. Taking n = 0 in series definition (2.4) of the Sheffer–Appell polynomials and then using Eq. (2.9) in the resultant equation, we get assertion (2.7). In order to prove assertion (2.8), we expand the determinant given in Eq. (2.10) with respect to the first row, so that we have
S. Khan, M. Riyasat / J. Math. Anal. Appl. 421 (2015) 806–829
An (x) =
(−1)n (β0 )n+1
β1 β0 0 . . 0
β2
2 1 β1 β0
···
.
···
.
0
···
β0
+
···
.
.
···
.
0
···
β0
β0 0 0 . . 0
···
0
βn−1
n−1 ··· 1 βn−2
n−1 ··· 2 βn−3
.
···
.
.
···
.
0
···
β0
β0
β0 0 0 . . 0
(−1) x (β0 )n+1
x (β0 )n+1
.
β1
···
β0
β2
2 1 β1
0
β0
···
.
.
···
β1
···
.
.
···
0
0
···
···
β0
β2
2 1 β1
0
β0
···
.
.
···
.
.
···
0
0
···
β1
βn
n 1 βn−1
n 2 βn−2 . .
n n−1 β1
βn−1
n−1 ··· 1 βn−2
n−1 ··· 2 βn−3
β0
2n−1 n−1
.
···
β2
2 1 β1
β0 0 (−1)n x2 0 + (β0 )n+1 . . 0
n
βn−1
n−1 ··· 1 βn−2
n−1 ··· 2 βn−3
.
β0 0 n 0 (−1) x − (β0 )n+1 . . 0
+
···
···
813
βn
n β n−1
n1 2 βn−2 . .
n n−1 β1 βn
n 1 βn−1
n 2 βn−2 + ··· . .
n n−1 β1 βn
n 1 βn−1
n 2 βn−2 . .
n n−1 β1
βn−1
n−1 1 βn−2
n−1 2 βn−3 . . . β0
(2.11)
ˆ s in Eq. (2.11) and then Since each minor in Eq. (2.11) is independent of x, therefore replacing x by M using the fact that s0 (x) = 1 and sn (x) = Msn {s0 (x)} (n = 1, 2, 3, · · ·) in the r.h.s. of the resultant equation, we find
n ˆ s ) = (−1) An (M (β0 )n+1
β1 β0 0 . . 0
β2
2 1 β1 β0
···
βn−1
n−1 ··· 1 βn−2
n−1 ··· 2 βn−3
.
···
.
.
···
.
0
···
β0
βn
n 1 βn−1
n 2 βn−2 . .
n n−1 β1
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β0 0 (−1)n s1 (x) 0 − (β0 )n+1 . . 0 β0 0 n (−1) s2 (x) 0 + (β0 )n+1 . . 0
βn
n 1 βn−1
n β0 2 βn−2 . ··· . . . ··· . .
n 0 ··· β0 n−1 β1 β1 · · · βn−1 βn
n β0 · · · n−1 β β n−2 n−1 1
n−1
n1 0 ··· 2 βn−3 2 βn−2 + ··· . ··· . . . ··· . .
n 0 ··· β0 n−1 β1 β0 β1 β2 ··· βn
2 0 β · · · n1 βn−1 0 1 β1
(−1)2n−1 sn−1 (x) 0 0 β0 · · · n2 βn−2 + (β0 )n+1 . . ··· . . . . . ··· .
n 0 0 0 ··· n−1 β1 β2 ··· βn−1 β0 β1
2 n−1 0 β0 1 β1 · · · 1 βn−2
n−1 0 β0 ··· sn (x) 0 2 βn−3 + . n+1 (β0 ) . . ··· . . . . . · · · .
n 0 0 0 ··· β 1 n−1 β2
2 1 β1
···
βn−1
n−1 ··· 1 βn−2
n−1 ··· 2 βn−3
(2.12)
Now, using operational rule (2.6) in the l.h.s. and combining the terms in the r.h.s. of Eq. (2.12), we get assertion (2.8). 2 The Appell and Sheffer polynomials, An (x) and sn (x) respectively, are quasi-monomial. In order to show that the Sheffer–Appell polynomials s An (x) are quasi-monomial, we prove the following result. Theorem 2.4. The Sheffer–Appell polynomials s An (x) are quasi-monomial with respect to the following multiplicative and derivative operators: −1 −1
ˆ sA = xH H −1 (Dx ) + A (H (Dx )) + A (H (Dx )) , M −1 −1 A(H (Dx )) A(H (Dx ))
(2.13a)
or, equivalently ˆ sA = M
g (f (Dx )) g (Dx ) 1 − x− g(Dx ) f (Dx ) g(f (Dx ))
(2.13b)
and PˆsA = H −1 (Dx ),
(2.14a)
S. Khan, M. Riyasat / J. Math. Anal. Appl. 421 (2015) 806–829
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or, equivalently PˆsA = f (Dx ),
(2.14b)
respectively. Proof. Differentiating Eq. (2.1a) partially with respect to t, we find
xH (t) +
∞ A (t) A (t) tn + A(t)A(t)exH(t) = . s An+1 (x) A(t) A(t) n! n=0
(2.15)
(t) A (t) Since A(t) and A(t) are invertible series of t, therefore A A(t) and A(t) possess power series expansions of t. Thus, in view of the following identity for the Sheffer–Appell polynomials:
Dx A(t)A(t)exH(t) = H(t)A(t)A(t)exH(t) ,
(2.16a)
H −1 (Dx ) A(t)A(t)exH(t) = tA(t)A(t)exH(t) ,
(2.16b)
or, equivalently
Eq. (2.15) becomes
xH H
−1
∞ A (H −1 (Dx )) A (H −1 (Dx )) tn xH(t) + A(t)A(t)e . (Dx ) + = A (x) n+1 s A(H −1 (Dx )) A(H −1 (Dx )) n! n=0
(2.17)
Again, using generating function (2.1a) in the l.h.s. of Eq. (2.17) and rearranging the summation, we have ∞ n=0
A (H −1 (Dx )) A (H −1 (Dx )) xH H −1 (Dx ) + + A(H −1 (Dx )) A(H −1 (Dx ))
s
tn An (x) n!
=
∞
s An+1 (x)
n=0
tn . n!
Equating the coefficients of the same powers of t in both sides of the above equation, we find
A (H −1 (Dx )) A (H −1 (Dx )) s xH H −1 (Dx ) + + An (x) = s An+1 (x), −1 −1 A(H (Dx )) A(H (Dx ))
(2.18)
which in view of monomiality principle equation (1.16) yields assertion (2.13a). Also, in view of relations (1.8), (1.14) and (1.15), Eq. (2.13a) can be expressed equivalently as (2.13b). In order to prove assertion (2.14a), we use generating function (2.1a) in both sides of the identity (2.16b), so that we have ∞ ∞ tn tn −1 = . (2.19) H (Dx ) s An (x) s An−1 (x) n! (n − 1)! n=0 n=1 Rearranging the summation in the l.h.s. of Eq. (2.19) and then equating the coefficients of the same powers of t in both sides of the resultant equation, we find s
H −1 (Dx )
An (x) = ns An−1 (x),
n ≥ 1,
(2.20)
which in view of monomiality principle equation (1.17) yields assertion (2.14a). Also, in view of relation (1.15), Eq. (2.14a) can be expressed equivalently as (2.14b). 2
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Remark 2.2. We remark that Eqs. (2.18) and (2.20) are the differential recurrence relations satisfied by the Sheffer–Appell polynomials s An (x). Theorem 2.5. The Sheffer–Appell polynomials s An (x) satisfy the following differential equation:
−1 −1 A (H −1 (Dx )) −1 A (H −1 (Dx )) −1 xH H (Dx ) H (Dx ) + H (Dx ) + H (Dx ) − n s An (x) = 0, A(H −1 (Dx )) A(H −1 (Dx ))
(2.21a)
or, equivalently g (Dx ) f (Dx ) g (f (Dx )) x− − f (D ) − n x s An (x) = 0. g(Dx ) f (Dx ) g(f (Dx ))
(2.21b)
Proof. Using Eqs. (2.13a) (or (2.13b)) and (2.14a) (or (2.14b)) in monomiality principle equation (1.19), we get assertion (2.21a) (or (2.21b)). 2 Further, we remark that operational rule (2.6) can be used to derive the results for the Sheffer–Appell polynomials from the results of the Appell polynomials. Several identities involving Appell polynomials are known. To give an example, we consider the following results for the Appell polynomials An (x) [9, (31)–(32) p. 1534]:
n−1 n 1 An (x) = xn − βn−k Ak (x) , n = 1, 2, · · · , β0 k k=0 n n n x = βn−k Ak (x), n = 0, 1, · · · . k
(2.22)
(2.23)
k=0
ˆ s in Eqs. (2.22), (2.23) and then using operational rules (2.6) and in view of Eq. (1.20), Replacing x by M we find the following results for the Sheffer–Appell polynomials s An (x):
n−1 n 1 sn (x) − βn−ks Ak (x) , n = 1, 2, · · · , s An (x) = β0 k k=0 n n sn (x) = βn−ks Ak (x), n = 0, 1, · · · . k
(2.24)
(2.25)
k=0
It is important to note that the Sheffer–Appell polynomials introduced in this section are actually the Sheffer polynomials, since their generating function is of the type A(t)exH(t) , with a suitable choice for A (t). In the next section, the Sheffer–Bernoulli and Sheffer–Euler polynomials are introduced as special cases of the Sheffer–Appell polynomials s An (x). 3. Sheffer–Bernoulli and Sheffer–Euler polynomials The typical examples of Appell polynomials besides the trivial example {xn }∞ n=0 are the Bernoulli, Euler and Hermite polynomials. In particular, Hermite polynomials sequence Hen (x) defined by the generating function t2
e(xt− 2 ) =
∞ n=0
Hen (x)
tn , n!
|t| < ∞; |x| < ∞
(3.1)
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817
Table 1 Some known Appell polynomials. S. No.
g(t); A(t) g(t) =
(et −1) ; t
g(t) =
(et +1) ; 2
III.
g(t) =
(e −1) tα
IV.
g(t) =
(et +1)α 2α
V.
g(t) =
I. II.
A(t) =
t
α
A(t) =
t (et −1)
Generating functions xt t tn = ∞ n=0 Bn (x) n! (et −1) e
A(t) =
2 (et +1)
xt 2 (et +1) e
t (et −1)α
t (et −1)α
ext =
; A(t) =
2α (et +1)α
2α (et +1)α
ext =
∞
(eα1 t −1)(eα2 t −1)...(eαm t −1) ; α1 α2 ...αm tm α1 α2 ...αm tm (eα1 t −1)(eα2 t −1)...(eαm t −1) α1 t
=
tn n=0 En (x) n!
; A(t) =
α
α2 t
αm t
VI.
+1) g(t) = (e +1)(e 2+1)...(e ; m m A(t) = (eα1 t +1)(eα2 t2+1)...(eαm t +1)
VII.
g(t) = A(t) =
h et − m−1 (t ) h=0 h! ; tm tm th et − m−1 ( ) h=0
The generalized Bernoulli polynomials [16]
n
n=0
(α) En (x) tn!
The generalized Euler polynomials [16]
α1 α2 ...αm tm xt (eα1 t −1)(eα2 t −1)...(eαm t −1) e ∞ n (m) = n=0 Bn (x|α1 , α2 , · · · , αm ) tn!
The generalized Bernoulli polynomials of order m [15]
xt 2m (eα1 t +1)(eα2 t +1)...(eαm t +1) e ∞ n (m) = n=0 En (x|α1 , α2 , · · · , αm ) tn!
The generalized Euler polynomials of order m [15]
h!
IX.
g(t) =
X.
g(t) = ( λet2+1 )−α ; A(t) = ( λet2+1 )α
XI.
g(t) =
XII.
g(t) = e−(ξ◦ +ξ1 t+ξ2 t +···+ξr+1 t ) ; 2 r+1 A(t) = e(ξ◦ +ξ1 t+ξ2 t +···+ξr+1 t )
=
∞ n=0
XIII.
g(t) = (1 − t)m+1 ; A(t) =
XIV.
g(t) =
∞
( λett−1 )α ext =
A(t) =
A(t) =
A(t) =
xt t λ et −1 e
t λet −1
2 λet +1
2
(et +1) ; 2t
(α) Bn (x) tn!
n
[m−1] Bn (x) tn!
h!
g(t) = ( λett−1 )−α ; A(t) = ( λett−1 )α
λet +1 ; 2
The Euler polynomials [25] n
n=0
tm ext h et − m−1 (t ) h=0
VIII.
λet −1 ; t
The Bernoulli polynomials [25]
∞
∞
α
Polynomials
r+1
1 (1−t)m+1
2t (et +1)
=
n
n=0
∞
t B(α) n (x; λ) n!
n
Bn (x; λ) tn!
n=0
(α) tn ( λet2+1 )α ext = ∞ n=0 En (x; λ) n! α |t + log λ| < π; 1 := 1 xt 2 tn = ∞ n=0 En (x; λ) n! λet +1 e |t + log λ| < π 2
1 t+ξ2 t +···+ξr+1 t e(ξ◦ +ξ tn = ∞ n=0 An (x) n!
1 (1−t)m+1
ext =
xt 2t (et +1) e
=
∞ n=0
∞ n=0
r+1
) xt
e
G(m) (x)tn , n
n
Gn (x) tn! ,
The new generalized Bernoulli polynomials [7] The Apostol–Bernoulli polynomials of order α [22] The Apostol–Bernoulli polynomials [2,22] The Apostol–Euler polynomials of order α [21,22] The Apostol–Euler polynomials [2,21,22] The generalized Gould–Hopper polynomials [14] (For r = 1, the Hermite polynomials Hn (x) [1] and for r = 2, classical 2-orthogonal polynomials) The Miller–Lee polynomials [1,12] (For m = 0, the truncated exponential polynomials en (x) [1] and for m = β − 1, the modified Laguerre polynomials (β) fn (x) [23]) The Genocchi polynomials [13]
is a unique sequence of Appell polynomials that is also orthogonal with respect to a positive measure. Some known Appell polynomials are listed in Table 1. The Bernoulli polynomials Bn (x) and Euler polynomials En (x) are important members of the Appell family. Therefore, in this section, we introduce the Sheffer–Bernoulli and Sheffer–Euler polynomials, s Bn (x) and s En (x) respectively, by means of generating functions and series definitions. The determinantal definitions for these polynomials are also given. 1 t Taking A(t) (or g(t) ) = et −1 of the Bernoulli polynomials Bn (x) in the l.h.s. of generating function (2.1a) (or (2.1b)) and denoting the resultant Sheffer–Bernoulli polynomials in the r.h.s. by sBn (x), we get the following generating function for the Sheffer–Bernoulli polynomials s Bn (x): ∞ tn t xH(t) A(t)e , = s Bn (x) t e −1 n! n=0
(3.2a)
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S. Khan, M. Riyasat / J. Math. Anal. Appl. 421 (2015) 806–829
or, equivalently ∞ 1 tn t x(f −1 (t)) e . = s Bn (x) t −1 e − 1 g(f (t)) n! n=0
(3.2b)
1 Similarly, taking A(t) (or g(t) ) = et2+1 of the Euler polynomials En (x) in the l.h.s. of generating function (2.1a) (or (2.1b)) and denoting the resultant Sheffer–Euler polynomials in the r.h.s. by s En (x), we get the following generating function for the Sheffer–Euler polynomials s En (x): ∞ tn 2 xH(t) A(t)e , = E (x) n s et + 1 n! n=0
(3.3a)
∞ 1 tn 2 x(f −1 (t)) e . = E (x) n s et + 1 g(f −1 (t)) n! n=0
(3.3b)
or, equivalently
Now, taking An (x) = Bn (x) in the l.h.s. and Ak = Bk in the r.h.s. of Eq. (2.4), we get the following series definition for the Sheffer–Bernoulli polynomials s Bn (x):
s Bn (x) =
n n Bk sn−k (x). k
(3.4)
k=0
Similarly, for An (x) = En (x) and Ak = Ek , we get the following series definition for the Sheffer–Euler polynomials s En (x):
s En (x) =
n n Ek sn−k (x). k
(3.5)
k=0
Remark 3.1. Next, taking An (x) = Bn (x) in Eq. (2.6), we get the following relation between the Bernoulli and Sheffer–Bernoulli polynomials: s Bn (x)
ˆ s ). = Bn (M
(3.6)
Similarly, for An (x) = En (x), we get the following relation between the Euler and Sheffer–Euler polynomials: s En (x)
ˆ s ). = En (M
(3.7)
1 It has been shown in [9] that for β0 = 1 and βi = i+1 (i = 1, 2, · · · , n) the determinantal definition of the Appell polynomials An (x) given by Eqs. (2.9) and (2.10) reduces to the determinantal form of the Bernoulli polynomials Bn (x) [8]. 1 Therefore, taking β0 = 1 and βi = i+1 (i = 1, 2, · · · , n) in Eqs. (2.7) and (2.8), we get the following determinantal definition of the Sheffer–Bernoulli polynomials s Bn (x):
S. Khan, M. Riyasat / J. Math. Anal. Appl. 421 (2015) 806–829
819
Definition 3.1. The Sheffer–Bernoulli polynomials s Bn (x) of degree n are defined by s B0 (x)
= 1,
1 s1 (x) s2 (x) 1 1 1 2
23 1 0 1 1 2 n 0 1 s Bn (x) = (−1) 0 . . . . . . 0 0 0
··· ··· ··· ···
sn−1 (x) 1
n−1n
1
1
n−1 n−1 1 2
n−2
···
.
···
.
···
1
sn (x) 1
n+1 n 1 1 n
n 1 , 2 n−1 . .
n 1
(3.8)
n = 1, 2, · · · ,
(3.9)
n−1 2
where sn (x) (n = 0, 1, 2, · · ·) are the Sheffer polynomials of degree n. Further, in view of the fact that for β0 = 1 and βi = 12 (i = 1, 2, · · · , n), Eqs. (2.9) and (2.10) give the determinantal form of the Euler polynomials [9, (60)–(61) p. 1540], so we get the following determinantal definition of the Sheffer–Euler polynomials s En (x): Definition 3.2. The Sheffer–Euler polynomials s En (x) of degree n are defined by s E0 (x)
= 1,
1 s1 (x) 1 1 2 0 1 n 0 s En (x) = (−1) 0 . . . . 0 0
s2 (x) · · · sn−1 (x) 1
2 1 2
···
1
2 1 n−1
2 1
···
1
···
2
1 1 n−1 2 2
.
···
.
.
···
.
0
···
1
sn (x) 1 2
1 n 2 1
1 n , 2 2 . .
n 1
(3.10)
n = 1, 2, · · · ,
(3.11)
2 n−1
where sn (x) (n = 0, 1, 2, · · ·) are the Sheffer polynomials of degree n. In order to frame the Sheffer–Bernoulli and Sheffer–Euler polynomials, sBn (x) and s En (x) respectively, within the context of monomiality principle, we prove the following results: Theorem 3.1. The Sheffer–Bernoulli polynomials s Bn (x) are quasi-monomial with respect to the following multiplicative and derivative operators: −1
−1 (H (Dx ))
(1 − (H −1 (Dx ))) − 1 ˆ sB = xH H −1 (Dx ) + A (H (Dx )) + e , M A(H −1 (Dx )) H −1 (Dx )(e(H −1 (Dx )) − 1)
(3.12a)
or, equivalently ˆ sB = M
1 ef (Dx ) (1 − f (Dx )) − 1 g (Dx ) + x− g(Dx ) f (Dx ) f (Dx )(ef (Dx ) − 1)
(3.12b)
and PˆsB = H −1 (Dx ),
(3.13a)
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820
or, equivalently PˆsB = f (Dx ),
(3.13b)
respectively. Proof. For the Bernoulli polynomials Bn (x), we have A(t) (or −1
(H (Dx )) e(H −1 (Dx )) −1
and consequently A (H −1 (Dx )) =
e
(H −1 (Dx ))
1 t g(t) ) = et −1 .
Therefore, taking A(H −1 (Dx )) =
−1
(1−(H (Dx )))−1 (e(H −1 (Dx )) −1)2
in Eq. (2.13a), we get assertion
(3.12a). (f (Dx ))
(f (Dx )−1)+1 Similarly, taking g(f (Dx )) = e f (Dx−1 and consequently g (f (Dx )) = e in Eq. (2.13b), ) (f (Dx ))2 we get assertion (3.12b). Further, in view of Eq. (2.14a) (or (2.14b)), we get assertion (3.13a) (or (3.13b)). 2 f (Dx )
Theorem 3.2. The Sheffer–Bernoulli polynomials s Bn (x) satisfy the following differential equation:
A (H −1 (Dx )) −1 xH H −1 (Dx ) H −1 (Dx ) + H (Dx ) A(H −1 (Dx )) −1 e(H (Dx )) (1 − (H −1 (Dx ))) − 1 −1 + H (Dx ) − n s Bn (x) = 0, H −1 (Dx )(e(H −1 (Dx )) − 1)
(3.14a)
or, equivalently g (Dx ) f (Dx ) ef (Dx ) (1 − f (Dx )) − 1 f (Dx ) − n s Bn (x) = 0. x− + g(Dx ) f (Dx ) f (Dx )(ef (Dx ) − 1)
(3.14b)
Proof. Using Eqs. (3.12a) (or (3.12b)) and (3.13a) (or (3.13b)) in monomiality principle equation (1.19), we get assertion (3.14a) (or (3.14b)). 2 Theorem 3.3. The Sheffer–Euler polynomials s En (x) are quasi-monomial with respect to the following multiplicative and derivative operators: −1
−1 (H (Dx ))
ˆ sE = xH H −1 (Dx ) + A (H (Dx )) − e −1 , M −1 H (Dx ) + 1) A(H (Dx )) (e
(3.15a)
or, equivalently ˆ sE = M
x−
1 ef (Dx ) g (Dx ) − f (D ) g(Dx ) f (Dx ) (e x + 1)
(3.15b)
PˆsE = H −1 (Dx ),
(3.16a)
PˆsB = f (Dx ),
(3.16b)
and
or, equivalently
respectively.
S. Khan, M. Riyasat / J. Math. Anal. Appl. 421 (2015) 806–829
Proof. For the Euler polynomials En (x), we have A(t) (or 2 e(H −1 (Dx )) +1
and consequently A (H −1 (Dx )) = −
1 g(t) )
−1 2e(H (Dx )) (e(H −1 (Dx )) +1)2
=
2 et +1 .
821
Therefore, taking A(H −1 (Dx )) =
in Eq. (2.13a), we get assertion (3.15a).
f (Dx )
f (Dx )
Similarly, taking g(f (Dx )) = e 2 +1 and consequently g (f (Dx )) = e 2 in Eq. (2.13b), we get assertion (3.15b). Further, in view of Eq. (2.14a) (or (2.14b)), we get assertion (3.16a) (or (3.16b)). 2 Theorem 3.4. The Sheffer–Euler polynomials s En (x) satisfy the following differential equation:
−1
A (H −1 (Dx )) −1 e(H (Dx )) −1 xH H −1 (Dx ) H −1 (Dx ) + H H (D ) − (D ) − n x x s En (x) = 0, A(H −1 (Dx )) (eH −1 (Dx ) + 1) (3.17a)
or, equivalently g (Dx ) f (Dx ) ef (Dx ) − f (Dx ) − n s En (x) = 0. x− g(Dx ) f (Dx ) ef (Dx ) + 1
(3.17b)
Proof. Using Eqs. (3.15a) (or (3.15b)) and (3.16a) (or (3.16b)) in monomiality principle equation (1.19), we get assertion (3.17a) (or (3.17b)). 2 Next, we consider the applications of operational rules (3.6) and (3.7) to derive the results for the Sheffer–Bernoulli and the Sheffer–Euler polynomials, from the results of the Bernoulli and Euler polynomials respectively. We recall the following functional equations involving the Bernoulli and Euler polynomials: n−1 k=0
n Bk (x) = nxn−1 , k
n = 2, 3, · · · ,
n−1 1 n En (x) = x − n Ek (x), 2 k n
n = 1, 2, · · · .
(3.18)
(3.19)
k=0
ˆ s of the Sheffer polynomials sn (x) and then using apReplacing x by the multiplicative operator M propriate operational rules in the resultant equations, we find the following functional equations for the Sheffer–Bernoulli and Sheffer–Euler polynomials: n−1 k=0
n s Bk (x) = nsn−1 (x), k
n−1 1 n s En (x) = sn (x) − n s Ek (x), 2 k
n = 2, 3, · · · ,
(3.20)
n = 1, 2, · · · ,
(3.21)
k=0
respectively. The above examples illustrate that the operational relations established in this paper can be used to derive the results for the newly introduced families of polynomials from the results of any member belonging to the Appell family.
S. Khan, M. Riyasat / J. Math. Anal. Appl. 421 (2015) 806–829
822
Table 2 Some known Sheffer polynomials. S. No.
A(t); H(t) −t2
g(t); f (t)
I.
e
; 2t
e
II.
e−t ; νt
III.
(1 − t)−1 ;
IV.
(1 − t)−α−1 ;
t2 4
;
t
m
m
e( ν ) ; t t−1
t t−1
1+t ln( 1−t )
Generating functions 2 tn e2xt−t = ∞ n=0 Hn (x) n!
t 2
exp(νxt − tm ) =
t ν
(1 − t)−1 ;
t t−1
exp(βt + x(1 − et )) =
exp(a(et −1)); a(et − 1)
e−t (1 +
λ −μ x (1 + (1 + t) ) (1 + t)tn = ∞ s (x; λ, μ) n n=0 n! ∞ x t tn n=0 bn (x) n! ln(1+t) (1 + t) =
e−t ; ln(1 +
VIII.
(1 + (1 + t)λ )−μ ; ln(1 + t)
(1 + eλt )μ ; et − 1
IX.
t ln(1+t) ;
t et −1 ;
X.
2 2+t ;
XI.
1 1+t2
XII.
(1 − 4t) 2 × ( 1+√21−4t )a−1 ; (1+
−4t √ 1−4t)2
∞
et − 1
xt exp ( t−1 )=
=
t x a)
∞
=
n=0
n=0
Laguerre polynomials n!Ln (x) [1]
n L(α) n (x)t
Generalized Laguerre polynomials n!L(α) n (x) [1,25]
n
Pn (x) tn!
∞
n
n=0
t a(β) n (x) n!
Actuarial polynomials a(β) n (x) [6]
cn (x; a) tn!
+ et ); e −1
2 2+t (1
+ t)x =
sec t; tan t
1 1+t2
exp(x arctan(t)) =
n=0
Pidduck polynomials Pn (x) [6,16]
n
n=0
∞
∞
1 2 (1 t
1+t (1−t)a ; 1 1 1+t 2 4 − 4 ( 1−t )
Generalized Hermite polynomials Hn,m,ν (x) [20]
Ln (x)tn
(1 − t)−β ; ln(1 − t)
VII.
−1
n=0
1+t x t 1−t ( 1−t )
eβt ; 1 − et
; arctan(t)
∞
et −1 2 et −1 ; et +1
VI.
ln(1 + t)
Hn,m,ν (x) tn!
1 (1−t)α+1
t t−1
t 1−t ;
ln(1 + t)
n
n=0
xt exp ( t−1 )=
Hermite polynomials Hn (x) [1]
(1 − t)−α−1 ;
V.
t a)
1 (1−t)
∞
Polynomials
Poisson–Charlier polynomials cn (x; a) [15,17,29] Peters polynomials sn (x; λ, μ) [6] Bernoulli polynomials of the second kind bn (x) [17]
n
rn (x) tn! ∞ n=0
−1
Related polynomials rn (x) [17] n
Rn (x) tn!
√ (1 − 4t) 2 ( 1+√21−4t )a−1 exp( (1+−4xt ) 1−4t)2 n R (a, x)t = ∞ n n=0
Hahn polynomials Rn (x) [4] Shively’s pseudo-Laguerre polynomials Rn (a, x) [25]
Acknowledgments The authors are thankful to the reviewer(s) for several useful comments and suggestions towards the improvement of this paper. The authors would also like to express their thanks to the editor for encouraging comments. Appendix A The Appell and Sheffer polynomial sequences arise in numerous problems of applied mathematics, theoretical physics, approximation theory and several other mathematical branches. A list of polynomials belonging to the Appell family is given in Table 1. We present the list of some known Sheffer polynomials in Table 2. The Hermite and Laguerre polynomials are important members of the Sheffer family. Taking A(t) (or 1 −t2 and H(t) (or f −1 (t)) = 2t of the Hermite polynomials Hn (x) (Table 2(I)) in Eq. (2.1a) (or g(f −1 (t)) ) = e (2.1b)), we get the following generating function for the Hermite–Appell polynomials [19]: ∞
tn , n!
(A.1a)
∞ tn 1 (2xt−t2 ) e . = H An (x) g(t) n! n=0
(A.1b)
2
A(t)e(2xt−t
)
=
n=0
H An (x)
or, equivalently
S. Khan, M. Riyasat / J. Math. Anal. Appl. 421 (2015) 806–829
Table 3 Results for
H An (x)
and
823
L An (x).
S. No.
Results
I.
Series definitions
H An (x)
Hermite–Appell polynomials H An (x)
n = n k=0 k Ak Hn−k (x)
II.
Multiplicative and derivative operators
ˆ HA = 2x − Dx + M PˆHA =
Laguerre–Appell polynomials L An (x) n n L An (x) = k=0 k Ak Ln−k (x)
A (Dx /2) A(Dx /2)
ˆ LA = −xD 2 + (2x − 1)Dx − (x − 1) M x (Dx /Dx −1) + A A(Dx /Dx −1) PˆLA = Dx
Dx 2
Dx −1
III.
Differential equations
(Dx /2) 2 (Dx − 2xDx − A A(Dx /2) Dx + 2n)H An (x) = 0
2 (xDx − (2x − 1)Dx + (x − 1) (Dx /Dx −1) Dx −1 − A A(Dx /Dx −1) + n Dx )L An (x) = 0
IV.
Operational rules
ˆ H An (x) = An (MH ); ˆ H := Multiplicative operator of Hn (x) M
ˆ L An (x) = An (ML ); ˆ L := Multiplicative operator of Ln (x) M
1 1 t −1 Similarly, taking A(t) (or g(f −1 (t)) = t−1 of the Laguerre polynomials Ln (x) (t)) ) = 1−t and H(t) (or f (Table 2(III)) in Eq. (2.1a) (or (2.1b)), we get the following generating function for the Laguerre–Appell polynomials [18]: ∞ 1 ( −xt tn e 1−t ) = , L An (x) 1−t n! n=0
(A.2a)
∞ 1 ( −xt tn 1 e 1−t ) = . L An (x) g(t) 1 − t n! n=0
(A.2b)
A(t) or, equivalently
Certain results for H An (x) and L An (x), which are not considered in [19,18] are given in Table 3. Taking sn (x) = Hn (x) (n = 0, 1, · · ·) (Table 2(I)) in Eqs. (2.7) and (2.8), we get the following determinantal definition of the Hermite–Appell polynomials H An (x): Definition A.1. The Hermite–Appell polynomials H A0 (x)
=
1 , β0
β0 =
n
H An (x)
βn = −
=
1 A0
(−1)
n+1
(β0 )
1 β 0 0 0 . . 0
H An (x)
of degree n are defined by
1 , A0 H1 (x) β1
(A.3) H2 (x) · · ·
Hn−1 (x)
···
β0
β2
2 1 β1
0
β0
.
.
···
.
.
.
···
.
0
···
β0
0
n n Ak βn−k , k
βn−1
n−1 ··· 1 βn−2
n−1 ··· 2 βn−3
Hn (x) βn
n βn−1
n1 , 2 βn−2 . .
n n−1 β1
n = 1, 2, 3, · · · ,
(A.4)
k=1
where β0 , β1 , · · · , βn ∈ R, β0 = 0 and Hn (x) (n = 0, 1, · · ·) are the Hermite polynomials of degree n. Similarly, taking sn (x) = Ln (x) (n = 0, 1, · · ·) (Table 2(III)) in Eqs. (2.7) and (2.8), we get the following determinantal definition of the Laguerre–Appell polynomials L An (x):
S. Khan, M. Riyasat / J. Math. Anal. Appl. 421 (2015) 806–829
824
Table 4 Results for
H Bn (x)
and
H En (x).
S. No.
Results
I.
Series definitions
H Bn (x)
II.
Multiplicative and derivative operators
ˆ HB = 2x − Dx + M
Differential equations
2 (Dx − 2xDx −
Operational rules
+ 2n)H Bn (x) = 0 ˆ H Bn (x) = Bn (MH ); ˆ H := Multiplicative operator of Hn (x) M
III. IV.
Hermite–Bernoulli polynomials H Bn (x)
n = n k=0 k Bk Hn−k (x)
PˆHB =
eDx /2 (1− Dx )−1
ˆ HE = 2x − Dx − M
2
Dx 2
Dx 2
(eDx /2 −1)
PˆHE =
eDx /2 (1− Dx )−1 2
Dx 2
Hermite–Euler polynomials H En (x) n n H En (x) = k=0 k Ek Hn−k (x)
(eDx /2 −1)
2 (Dx − 2xDx +
Dx
eDx /2 eDx /2 +1
Dx 2 eDx /2 eDx /2 +1
+ 2n)H En (x) = 0
ˆ H En (x) = En (MH ); ˆ H := Multiplicative operator of Hn (x) M
Definition A.2. The Laguerre–Appell polynomials L An (x) of degree n are defined by L A0 (x)
=
1 , β0
β0 =
n
L An (x)
βn = −
=
1 A0
(−1)
n+1
(β0 )
n k=1
1 β 0 0 0 . . 0
1 , A0
(A.5)
L1 (x) L2 (x) · · · β1
Ln−1 (x)
···
β0
β2
2 1 β1
0
β0
.
.
···
.
.
.
···
.
0
···
β0
0
n Ak βn−k , k
βn−1
n−1 ··· 1 βn−2
n−1 ··· 2 βn−3
Ln (x) βn
n 1 βn−1
n , 2 βn−2 . .
n n−1 β1
n = 1, 2, 3, · · · ,
(A.6)
where β0 , β1 , · · · , βn ∈ R, β0 = 0 and Ln (x) (n = 0, 1, · · ·) are the Laguerre polynomials of degree n. With the help of the results given in Table 3 and by taking A(t) (or g(t)) of the members belonging to the Appell polynomials family, we can derive the generating functions and other results for the corresponding members of the Hermite–Appell and Laguerre–Appell families. 1 t To give an example, we take An (x) = Bn (x) then from Table 1(I), we have A(t) (or g(t) ) = et −1 and thus in view of Eq. (A.1a) (or (A.1b)), we get the following generating function for the Hermite–Bernoulli polynomials H Bn (x): ∞ tn t (2xt−t2 ) e . = H Bn (x) t e −1 n! n=0
(A.7)
1 Similarly, taking An (x) = En (x) (Table 1(II)), we have A(t) (or g(t) ) = et2+1 and thus in view of Eq. (A.1a) (or (A.1b)), we get the following generating function for the Hermite–Euler polynomials H En (x): ∞ 2 tn (2xt−t2 ) e . = H En (x) t e +1 n! n=0
(A.8)
The series definitions and other results for the Hermite–Bernoulli and Hermite–Euler polynomials are given in Table 4. 1 Taking β0 = 1 and βi = i+1 (i = 1, 2, · · · , n) in Eqs. (A.3) and (A.4), we find the following determinantal definition of the Hermite–Bernoulli polynomials H Bn (x):
S. Khan, M. Riyasat / J. Math. Anal. Appl. 421 (2015) 806–829
Definition A.3. The Hermite–Bernoulli polynomials H B0 (x)
H Bn (x)
of degree n are defined by
= 1,
1 1 0 n H Bn (x) = (−1) 0 . . 0
H1 (x) H2 (x) · · ·
Hn−1 (x)
···
1
1
23 1 1 2
0
1
···
.
.
···
.
.
.
···
.
0
0
···
1
1 2
···
825
1
n−1n
1
1
n−1 n−1 1 2
n−2
Hn (x) 1
n+1 n 1 1 n
n 1 , 2 n−1 . .
n 1
(A.9)
n = 1, 2, · · · ,
(A.10)
n−1 2
where Hn (x) (n = 0, 1, 2, · · ·) are the Hermite polynomials of degree n. Next, taking β0 = 1 and βi = 12 (i = 1, 2, · · · , n) in Eqs. (A.3) and (A.4), we find the following determinantal definition of the Hermite–Euler polynomials H En (x): Definition A.4. The Hermite–Euler polynomials H E0 (x)
H En (x)
of degree n are defined by
= 1,
1 1 0 n E (x) = (−1) 0 H n . . 0
H1 (x) H2 (x) · · · Hn−1 (x) 1 2
1
···
1
1
2 1 2
2 1 n−1
2 1
···
0
1
···
2
1 1 n−1 2 2
.
.
···
.
.
.
···
.
0
0
···
1
Hn (x) 1 2
1 n 2 1
1 n , 2 2 . .
n 1
(A.11)
n = 1, 2, · · · ,
(A.12)
2 n−1
where Hn (x) (n = 0, 1, 2, · · ·) are the Hermite polynomials of degree n. Also, in view of Eq. (A.2a) (or (A.2b)), we get the following generating functions for the Laguerre– Bernoulli and Laguerre–Euler polynomials L Bn (x) and L En (x): ∞ tn t ( −xt 1−t ) = e L Bn (x) t (e − 1)(1 − t) n! n=0
(A.13)
∞ tn 2 ( −xt 1−t ) = e , L En (x) t (e + 1)(1 − t) n! n=0
(A.14)
and
respectively. The series definitions and other results for the Laguerre–Bernoulli and Laguerre–Euler polynomials are given in Table 5. 1 Taking β0 = 1 and βi = i+1 (i = 1, 2, · · · , n) in Eqs. (A.5) and (A.6), we find the following determinantal definition of the Laguerre–Bernoulli polynomials L Bn (x):
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826
Table 5 Results for
L Bn (x)
and
L En (x).
S. No.
Results
I.
Series definitions
L Bn (x)
Laguerre–Bernoulli polynomials
n = n k=0 k Bk Ln−k (x)
II.
Multiplicative and derivative operators
ˆ LB = −xD 2 + (2x − 1)Dx − (x − 1) M x +
exp ( (
PˆLB = III.
Differential equations
Dx Dx −1
Dx Dx −1
L Bn (x)
Laguerre–Euler polynomials L En (x) n n L En (x) = k=0 k Ek Ln−k (x) ˆ LE = −xD 2 + (2x − 1)Dx − (x − 1) M x
Dx )−1 Dx −1 Dx )−1) Dx −1
)(1−
)(exp (
−
PˆLE =
Dx Dx −1
2 (xDx − (2x − 1)Dx + (x − 1)
−
+
Operational rules
ˆ L Bn (x) = Bn (ML ); ˆ L := Multiplicative operator of Ln (x) M
Dx ) Dx −1 Dx )+1) Dx −1
exp ( (exp (
x −1 + n DD )L Bn (x) = 0 x
IV.
Dx Dx −1
2 (xDx − (2x − 1)Dx + (x − 1)
Dx )(1− Dx )−1 Dx −1 Dx −1 Dx )(exp( Dx )−1) Dx −1 Dx −1
exp ( (
Dx ) Dx −1 Dx )+1) Dx −1
exp ( (exp (
x −1 + n DD )L En (x) = 0 x
ˆ L En (x) = En (ML ); ˆ L := Multiplicative operator of Ln (x) M
Definition A.5. The Laguerre–Bernoulli polynomials L Bn (x) of degree n are defined by L B0 (x)
= 1,
1 1 0 n B (x) = (−1) 0 L n . . 0
L1 (x) L2 (x)
···
Ln−1 (x)
···
1
1
23 1 1 2
0
1
···
.
.
···
.
.
.
···
.
0
0
···
1
1 2
···
1
n−1n
1
1
n−1 n−1 1 2
n−2
Ln (x) 1
n+1 n 1 1 n
n 1 , 2 n−1 . .
n 1
(A.15)
n = 1, 2, · · · ,
(A.16)
n−1 2
where Ln (x) (n = 0, 1, 2, · · ·) are the Laguerre polynomials of degree n. Taking β0 = 1 and βi = 12 (i = 1, 2, · · ·) in Eqs. (A.5) and (A.6), we find the following determinantal definition of the Laguerre–Euler polynomials L En (x): Definition A.6. The Laguerre–Euler polynomials L En (x) of degree n are defined by L E0 (x)
= 1,
1 1 0 n L En (x) = (−1) 0 . . 0
L1 (x) L2 (x)
· · · Ln−1 (x) ···
1
1 2
1 2 2 1
0
1
···
2
1 1 n−1 2 2
.
.
···
.
.
.
···
.
0
0
···
1
1 2
···
1
2 1 n−1
Ln (x) 1 2
1 n 2 1
n 1 , 2 2 . .
n 1
(A.17)
n = 1, 2, · · · ,
(A.18)
2 n−1
where Ln (x) (n = 0, 1, 2, · · ·) are the Laguerre polynomials of degree n. Further, we proceed to draw the graphs of H Bn (x), H En (x), L Bn (x) and L En (x). To draw the graphs of these polynomials, we find the values of the first five Hermite polynomials Hn (x) and Laguerre polynomials Ln (x) by making use of their generating functions given in Table 2(I) and (III). We give the list of first five Hermite and Laguerre polynomials in Table 6.
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827
Table 6 First five Hn (x) and Ln (x). n
0
1
2
3
4
Hn (x) Ln (x)
1 1
2x 1−x
4x2 − 2 2 1 2 (x − 4x + 2)
8x3 − 12x 3 2 1 6 (−x + 9x − 18x + 6)
16x4 − 48x2 + 12 4 3 2 1 24 (x − 16x + 72x − 96x + 24)
Next, we consider the values of H Bn (x), (A.10), (A.12), (A.16) and (A.18), we have
H En (x), L Bn (x)
and L En (x) for n = 4. Taking n = 4 in Eqs.
1 H1 (x) H2 (x) H3 (x) H4 (x) 1/2 1/3 1/4 1/5 1 1 1 1 1 , H B4 (x) = 0 0 1 3/2 2 0 0 0 0 1 2 1 H1 (x) H2 (x) H3 (x) H4 (x) 1/2 1/2 1/2 1/2 1 1 1 3/2 2 , H E4 (x) = 0 0 1 3/2 3 0 0 0 0 1 2 1 L1 (x) L2 (x) L3 (x) L4 (x) 1/3 1/4 1/5 1 1/2 1 1 1 1 L B4 (x) = 0 0 1 3/2 2 0 0 0 0 1 2
(A.19)
(A.20)
(A.21)
and 1 L1 (x) L2 (x) L3 (x) L4 (x) 1/2 1/2 1/2 1 1/2 1 1 3/2 2 , L E4 (x) = 0 0 1 3/2 3 0 0 0 0 1 2
(A.22)
respectively. Now, by using the expressions of first five Hn (x) and Ln (x) from Table 6 in Eqs. (A.19)–(A.22) and simplifying, we find
H B4 (x)
= 16x4 − 16x3 − 44x2 + 24x +
H E4 (x)
299 , 30
= 16x4 − 16x3 − 48x2 + 26x + 12, 1 4 1 3 1 2 1 x − x + x − , 24 3 2 30 1 4 1 3 x − x + x. L E4 (x) = 24 3
L B4 (x)
=
(A.23) (A.24) (A.25) (A.26)
828
S. Khan, M. Riyasat / J. Math. Anal. Appl. 421 (2015) 806–829
Using Eqs. (A.23)–(A.26), we get the following graphs:
We know that the Bernoulli numbers Bn [25] appear in Taylor series expansions of the tangent and hyperbolic tangent functions, while the Euler numbers En [25] appear in Taylor series expansions of the secant and hyperbolic secant functions. The Bernoulli and Euler numbers have deep connections with number theory and occur in combinatorics. These numbers also appear as special values of the Bernoulli and Euler polynomials. We note that Bn = Bn (0) and 1 En = 2 En 2 n
The importance of the Bernoulli and Euler numbers provides motivation to introduce the numbers related to the polynomial families introduced in this article. This aspect will be taken in next investigation. References [1] L.C. Andrews, Special Functions for Engineers and Applied Mathematicians, Macmillan Publishing Company, New York, 1985.
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