- Email: [email protected]
Differential and integral equations for the 2-iterated Appell polynomials
Journal of Computational and Applied Mathematics 306 (2016) 116–132
Contents lists available at ScienceDirect
Journal of Computational and Applied Mathematics journal homepage: www.elsevier.com/locate/cam
Differential and integral equations for the 2-iterated Appell polynomials Subuhi Khan ∗ , Mumtaz Riyasat Department of Mathematics, Aligarh Muslim University, Aligarh, India
article
abstract
info
Article history: Received 14 March 2015 Received in revised form 29 February 2016 MSC: 33E20 33E30 Keywords: Differential equations 2-iterated Appell polynomials Integral equations
In this article, a set of differential equations of finite order (kth order, k ∈ N) for the 2-iterated Appell polynomials are derived. Particular cases k = 1 and k = 2 are also considered. The integral equations for the Appell and 2-iterated Appell polynomials are established. Further, as an illustration the differential and integral equations for the 2-iterated generalized Bernoulli polynomials are derived. This article is first attempt in the direction of deriving integral equations for the Appell and 2-iterated Appell families and for some members belonging to these families. © 2016 Elsevier B.V. All rights reserved.
1. Introduction and preliminaries One of the important classes of polynomial sequences is the class of Appell polynomial sequences [1]. The Appell polynomial sequences appear in different applications in pure and applied mathematics. These 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 sequences. Properties of the Appell sequences are naturally handled within the framework of modern classical umbral calculus by Roman [2]. In 1936 an initial bibliography was provided by Davis [3]. Di Bucchianico [4] summarized and documented more than five hundred old and new findings related to the study of Appell sequences. The Appell polynomials Rn (x) [1] are defined by the following generating function: A(t )ext =
∞
Rn (x)
n =0
tn n!
,
(1.1)
where A(t ) has (at least the formal) expansion: A(t ) =
∞ n =0
Rn
tn n!
,
R0 := A(0) ̸= 0;
Rn := Rn (0).
(1.2)
It is easy to see that for any A(t ) the derivative of Rn (x) satisfies R′n (x) = n Rn−1 (x).
∗
Corresponding author. E-mail addresses: [email protected] (S. Khan), [email protected] (M. Riyasat).
http://dx.doi.org/10.1016/j.cam.2016.03.039 0377-0427/© 2016 Elsevier B.V. All rights reserved.
(1.3)
S. Khan, M. Riyasat / Journal of Computational and Applied Mathematics 306 (2016) 116–132
117
(α)
(α)
The Bernoulli and Euler polynomials Bn (x) and En (x) together with their familiar generalizations Bn (x) and En (x), respectively of (real or complex) order α [5,6] belong to the class of Appell sequences. t For A(t ) = et − , Eq. (1.1) gives the generating function for the Bernoulli polynomials [5]: 1 t
et − 1
∞
ext =
Bn (x)
n=0
tn n!
,
|t | < 2π .
(1.4)
The Bernoulli polynomials Bn (x) play an important role in various expansions and approximation formulas, which are useful both in analytic theory of numbers and in classical and numerical analysis. These polynomials provide solutions to various problems of engineering and physics. The Bernoulli polynomials are employed in the integral representation of differentiable periodic functions and play an important role in the approximation of such functions by means of polynomials. The Bernoulli numbers Bn := Bn (0) are defined by the following generating function [5]: t
et − 1
=
∞
Bn
n =0
tn n!
.
(1.5)
Moreover, we have 1
Bn (0) = (−1)n Bn (1) =
1
, n ∈ N0 . (1.6) 21−n − 1 2 The Bernoulli numbers Bn enter in many mathematical formulas, such as the Taylor expansion in a neighborhood of the origin of the trigonometric and hyperbolic tangent and cotangent functions [7], the sums of powers of natural numbers and residual term of the Euler Maclaurin quadrature formula [8]. Thus, the Bernoulli polynomials and Bernoulli numbers have deep connections with number theory and occur in combinatorics. (λ) Further, we recall that the generalized Bernoulli polynomials Bn (x) defined by the generating function [6]: t
et
λ
−1
ext =
∞
B(λ) n (x)
n =0
tn n!
Bn
,
|t | < 2π ,
belong to the class of Appell sequences with A(t ) =
(1.7)
t et −1
λ
(λ)
and generalized Bernoulli numbers Bn
:= B(λ) n (0) are defined
by the following generating function: t
et
λ
−1
=
∞
B(λ) n
n =0
tn n!
.
(1.8)
From Eqs. (1.4) and (1.7), it follows that B(n1) (x) = Bn (x).
(1.9)
Recently, Subuhi Khan and N. Raza [9] introduced the 2-iterated Appell polynomials by combining two different sets of Appell polynomials. We recall that the 2-iterated Appell polynomials (2IAP) R[n2] (x) are defined by the generating function [9, p. 9471(2.5)]: A1 (t ) A2 (t ) ext =
∞ n =0
R[n2] (x)
tn n!
,
(1.10)
where A1 (t ) =
∞
Rn
n =0
A2 (t ) =
∞
tn n!
Rn
n =0
,
tn n!
,
A1 (0) ̸= 0; A2 (0) ̸= 0;
Rn := Rn (0),
Rn := Rn (0).
(1.11)
(1.12)
The series definition for the 2IAP R[n2] (x) is given as [9, p. 9471(2.9)]: R[n2] (x) =
n n m=0
m
Rm Rn−m (x),
(1.13)
Rm Rn−m ,
(1.14)
where R[n2] (0) =
n n m=0
m
denotes the 2-iterated Appell numbers.
118
S. Khan, M. Riyasat / Journal of Computational and Applied Mathematics 306 (2016) 116–132
We recall that, if pn (x) and qn (x) = pn (x) is defined to be the sequence [2] qn (p(x)) =
n
n
k =0
qn,k xk are sequences of polynomials, then the umbral composition of qn (x) with
qn,k pk (x),
(1.15)
k =0
which is equivalent to condition (1.13). The set of all Appell sequences is closed under the operation of umbral compositions of polynomial sequences. Under this operation the set of all Appell sequences is an abelian group. Since the generating function of the 2IAP is of the form A⋆ (t )ext , with A⋆ (t ) as the product of two similar functions of t. Therefore, the set of all 2IAP sequences also form an abelian group under the operation of umbral composition. With the help of determinantal form of the 2IAP considered in [10], it may be possible to compute the coefficients or the value in a chosen point, for particular sequences of the 2IAP family, through an efficient and stable Gaussian algorithm. It can be useful in finding the solution of general linear interpolation problem. It has been shown in [9], that corresponding to each member belonging to the Appell family, there exists a new mixed special polynomial belonging to the 2IAP family. We recall that corresponding to the Bn (x) defined by Eq. (1.4), the 2-iterated Bernoulli polynomials (2IBP) B[n2] (x) are defined by the generating function [9, p. 9474(Table 2.1)]: t
et
−1
2
ext =
∞
B[n2] (x)
n =0
tn n!
.
(1.16) (λ)
Similarly, corresponding to the Bn (x) defined by Eq. (1.7), the 2-iterated generalized Bernoulli polynomials (2IGBP) (x) are defined by the generating function [9]: Bn (λ)[2]
t
2λ
et − 1
ext =
∞
2] B(λ)[ (x) n
n =0
tn n!
.
(1.17)
The recurrence relations are of fundamental importance in analysis of algorithms [11,12]. In digital signal processing, recurrence relations can model feedback in a system, where outputs at one time become inputs for future time. They thus arise in infinite impulse response (IIR) digital filters. The linear recurrence relations are also used extensively in both theoretical and empirical economics [13]. The study of differential equations is a wide field in pure and applied mathematics, physics, and engineering. The problems arising in different areas of science and engineering are usually expressed in terms of differential equations, which in most of the cases have special functions as their solutions. During the past three decades, the development of nonlinear analysis, dynamical systems and their applications to science and engineering has stimulated renewed enthusiasm for the theory of differential equations. Differential equations play an important role in modeling virtually every physical, technical, or biological process, from celestial motion, to bridge design, to interactions between neurons. Many fundamental laws of physics and chemistry can be formulated as differential equations. In biology and economics, differential equations are used to model the behavior of complex systems [14]. Sequences of Appell polynomials have been well studied because of their remarkable applications in mathematical and numerical analysis, as well as in number theory, as both classic literature [1,15,2,16,17] and more recent literature [18–25] testify. One aspect of such study for these sequences is to find differential equation and recurrence relations for Appell sequences. We recall some preliminaries from [24,26]. Let {pn (x)}∞ n=0 be a sequence of polynomials such that deg(pn (x)) = n, (n ∈ N0 := {0, 1, 2, . . .}). The differential operators Θn− and Θn+ satisfying the properties
Θn− {pn (x)} = pn−1 (x),
(1.18a)
Θn {pn (x)} = pn+1 (x),
(1.18b)
+
are called derivative and multiplicative operators, respectively. The monomiality principle [27,28] and the associated operational rules are used in [29] to explore new classes of isospectral problems leading to nontrivial generalizations of special functions. The polynomial sequence {pn (x)}∞ n=0 is called quasi-monomial, if and only if the above conditions are satisfied. Most of the properties of the families of polynomials associated with these operators can be deduced using operator rules with the Θn− and Θn+ operators. Obtaining the derivative and multiplicative operators of a given family of polynomials gives rise to some useful properties such as
(Θn−+1 Θn+ ){pn (x)} = pn (x),
(1.19a)
(Θn+−1 Θn+−2
(1.19b)
. . . Θ2 Θ1 Θ0 ){p0 (x)} = pn (x). +
+
+
Note that, if Θn− and Θn+ are differential realizations, then the above equations give the differential equation satisfied by pn (x). The technique used in obtaining differential equations via (1.19a) and (1.19b) is known as the factorization method.
S. Khan, M. Riyasat / Journal of Computational and Applied Mathematics 306 (2016) 116–132
119
The main idea of the factorization method is to find the derivative operator Θn− and multiplicative operator Θn+ such that Eq. (1.19a) holds. He and Ricci [24] used the classical factorization method introduced by Infeld and Hull [30] to study finite order differential equation for the Appell polynomials Rn (x). Motivated by this approach Özarslan and Yilmaz [26] extended this method to obtain a set of finite order (kth order, −(k) +(k) k ∈ N) differential equations for Rn (x) by constructing the operators Θn and Θn satisfying
Θn−(k) {pn (x)} = pn−k (x), +(k)
Θn
(1.20a)
{pn (x)} = pn+k (x), −(k)
(1.20b)
+(k)
the operators Θn and Θn are called k-times derivative and k-times multiplicative operators, respectively. Obtaining these operators for a given polynomial set will provide several useful relations for the corresponding polynomial set. Such −(k) +(k) as, when Θn and Θn are differential operators then, for each k ∈ N, the relation k) +(k) (Θn−( ){pn (x)} = pn (x), +k Θn
(1.21)
gives us the differential equations for this polynomial set. This method is called generalized factorization method [26] and it can be used to obtain a set of differential equations for pn (x), because for each k ∈ N, we have one differential equation for this polynomial. On the other hand, if n = mk + r then, by using few number of operators, Eq. (1.19b) can be expressed as +(k) +(k) +(k) + (Θn+−1 . . . Θmk Θ(m−1)k . . . Θk Θ0 ){p0 (x)} = pn (x).
(1.22)
Some problems which have their mathematical representation appear directly and in a natural way in terms of integral equations. Other problems, whose direct representations are in terms of differential equations and their auxiliary conditions may also be reduced to integral equations. Integral equations are important in many applications [31]. Problems in which integral equations encountered include radiative energy transfer and oscillations of a string, membrane or axle. Oscillation problems may also be solved as differential equations. In this article, recurrence relations and differential equations for the 2-iterated Appell polynomials are derived. This article is first attempt in the direction of introducing the integral equations for the Appell and 2-iterated Appell polynomials. +(k) −(k) (k ∈ N) are determined and a set of finite order differential equations for and Θn In Section 2, the shift operators Θn [2] the 2IAP Rn (x) is derived. The differential equations for k = 1 and k = 2 are also obtained as special cases. In Section 3, the integral equations for the Appell polynomials Rn (x) and 2IAP Rn[2] (x) are established. Finally, in Section 4, the shift operators, (λ)[2]
differential equations and integral equations for the 2IGBP Bn
(x) are derived.
2. Recurrence relations and differential equations −(k)
In this section, a set of finite order recurrence relation for the 2IAP R[n2] (x) is established and the shift operators Θn
and
+(k)
, for each k ∈ N are determined. Further, a set of finite order differential equations for the 2IAP R[n2] (x) is derived. In order to establish the recurrence relations, we prove the following result:
Θn
Theorem 2.1. For two different sets of Appell polynomials Rn (x) and Rn (x) and with A1 (t ) and A2 (t ) defined by Eqs. (1.11) and (1.12), let (m)
A1 (t ) A1 (t )
=
∞
αn(m)
n =0
tn
(2.1a)
n!
and (m)
A2 (t ) A2 (t )
=
∞
βn(m)
n =0
tn n!
,
(2.1b)
respectively. Then, the following finite set of recurrence relations for the 2IAP R[n2] (x) holds true: [2] Rn+k (x)
=
k k m=0
m
k−m
x
m m l=0
l
(l) (m−l) [2]
α0 β0
−l ) (l) (m−l) × (α0(l) βn(m )+ −s + αn−s β0
Rn (x) +
k k m=1
m
n n −s−1 n n−s s =0
s
p=1
p
x
k−m
m n −1 m n l =0
l
s=0
s
αn(l−) s−p βp(m−l) R[s2] (x).
(2.2)
120
S. Khan, M. Riyasat / Journal of Computational and Applied Mathematics 306 (2016) 116–132
Moreover, the k-times shift operators are given by n
Θn−(k) :=
Φm− =
m=n−k+1
n
1
m m=n−k+1
Dx =
(n − k)! k Dx n!
(2.3)
and +(k)
Θn
:=
k k m
m=0
x
k−m
m m l =0
−l ) × (α0(l) βn(m −s
+
l
(l) (m−l)
α0 β0
+
k k m=1
αn(l−) s β0(m−l) )Dnx −s
n
+
m
1
s =0
(n − s)!
x
k−m
m n −1 m l
l =0 n−s−1
n − s p
p=1
s=0
1
(n − s)!
αn(l−) s−p βp(m−l) Dnx −s
,
(2.4)
where Dx := ∂∂x . Proof. Let ∞
G(x, t ) = A1 (t )A2 (t )ext :=
R[n2] (x)
n=0
tn n!
.
(2.5)
Differentiating Eq. (2.5) k-times with respect to x and using the fact that
∂ kG = t k G(x, t ), ∂ xk we find ∞
R[n2] (x)
t n +k
n =0
n!
=
∞
Dkx {R[n2] (x)}
n=0
tn n!
,
which on equating the coefficients of same powers of t gives
(R[n2] )(k) (x) =
n!
Rn−k (x). [2]
(n − k)!
Since the operator Φn− =
1 D n x
(2.6)
satisfies the following operational relation [9]:
Φn− {R[n2] (x)} = R[n2−] 1 (x).
(2.7)
Therefore, considering the derivative operator as:
Φn− =
1 n
Dx
(2.8)
and in view of Eq. (2.6), we have
Θn−(k) {R[n2] (x)} :=
n
Φm− {R[n2] (x)} = R[n2−] k (x).
(2.9)
m=n−k+1
Using Eqs. (2.6) and (2.8) on the r.h.s. of Eq. (2.9), we find
Θn−(k) {R[n2] (x)} :=
n m=n−k+1
Φm− {R[n2] (x)} =
n
1
m m=n−k+1
Dx {R[n2] (x)} =
(n − k)! k [2] Dx {Rn (x)}, n!
(2.10)
which proves that the k-times derivative operator is given by (2.3). We differentiate the relation G(x, t ) = A1 (t )A2 (t )ext by using formula:
k ∂k k k−m { f ( t ) g ( t )} = Dm {g (t )}, t {f (t )}Dt ∂tk m m=0 so that we have
k ∂ k G(x, t ) k k−m xt = Dm {e } t {A1 (t )A2 (t )}Dt ∂tk m m=0
(2.11)
S. Khan, M. Riyasat / Journal of Computational and Applied Mathematics 306 (2016) 116–132
121
that is
k m ∂ k G(x, t ) k m l xt k−m = e x Dt {A1 (t )}Dtm−l {A2 (t )}, k ∂t m l m=0 l =0 which on multiplying and dividing by A1 (t )A2 (t ) gives
(l) (m−l) k m ∂ k G(x, t ) k m A1 (t ) A2 (t ) k−m = G ( x , t ) x . k ∂t m l A1 (t ) A2 (t ) m=0 l=0
(2.12)
Now, differentiating Eq. (2.5) k-times with respect to t and then using expression (2.12) in the resultant equation, we have G(x, t )
k k m
m=0
(l) (m−l) m m A (t ) A (t ) 1
xk−m
A1 (t )
l
l =0
2
=
A2 (t )
∞
Rn+k (x) [2]
n =0
tn n!
,
which on making use of Eqs. (2.5), (2.1a) and (2.1b) on the l.h.s. becomes ∞
R s ( x) [2]
s =0
k ts
s! m=0
m
k
x
k−m
m ∞ n m n l
l =0
p
n=0 p=0
t αn(l−) p βp(m−l)
n
n!
=
∞
Rn+k (x) [2]
n =0
tn n!
.
(2.13)
Regulating the series in Eq. (2.13) and then equating the coefficients of the same powers of t, we find [2] Rn+k (x)
=
k k m
m=0
x
k−m
m n n −s m n n − s l
l =0
s
s=0
p
p=0
αn(l−) s−p βp(m−l) Rs[2] (x),
or, equivalently Rn+k (x) = [2]
k k m
m=0
+
xk−m
l
l =0
k k
m=0
m n n−s−1 m n n − s
m
xk−m
s
s=0
p=0
m n m n
l
l =0
s=0
p
s
αn(l−) s−p βp(m−l) R[s2] (x)
−l) [2] α0(l) βn(m −s Rs (x).
(2.14)
Since 1, 0,
αn(0) = βn(0) = δn,0 :=
n=0 otherwise.
(2.15)
Therefore, we have [2] Rn+k (x)
=
k k m=0
×
m
k −m
x
m m l =0
n−1 n
s
s=0
l
(l) (m−l) [2]
α0 β0
Rn (x) +
k k m=1
−l) [2] α0(l) βn(m −s Rs (x)
+
m
xk−m
l =0
n n −s−1 n n−s s=0
s
p=0
m m
p
l
αn(l−) s−p βp(m−l) R[s2] (x)
or, equivalently [2] Rn+k (x)
=
k k m=0
×
m
x
n−1 n
s
n n s=0
m m l =0
s=0
+
k −m
s
l
(l) (m−l) [2]
α0 β0
Rn (x) +
k k m=1
−l) [2] α0(l) βn(m −s Rs (x) +
m
xk−m
n n −s−1 n n−s s=0
s
p=1
p
m m l =0
l
αn(l−) s−p βp(m−l) R[s2] (x)
αn(l−) s β0(m−l) R[s2] (x) ,
which on rearranging the summations in the second term of the r.h.s. yields assertion (2.2).
,
122
S. Khan, M. Riyasat / Journal of Computational and Applied Mathematics 306 (2016) 116–132
Further, in view of Eqs. (2.6) and (2.9), we have n
R[s2] (x) =
Φm− {R[n2] (x)} =
m=s+1
s! n!
Dnx −s [R[n2] (x)].
(2.16)
Using Eq. (2.16) in recurrence relation (2.2), we get
[2] Rn+k (x)
=
k k m
m=0
xk−m
m m
l
l =0
−l ) × (α0(l) βn(m −s
+
α0(l) β0(m−l) +
k k m
m=1
αn(l−) s β0(m−l) )Dnx −s
+
1
s=0
(n − s)! +(k)
m n −1 m
l
l=0
s=0
n−s−1
n
which proves that the k-times multiplicative operator Θn
xk−m
n − s
(n − s)!
αn(l−) s−p βp(m−l) , Dxn−s
p
p=1
1
is given by Eq. (2.4).
R[n2] (x)
(2.17)
Deriving a set finite order differential equations for the 2IAP Rn (x) is a new investigation. A set of finite order differential equations for the 2IAP R[n2] (x) for each k ∈ N is derived by proving the following result: [2]
Theorem 2.2. For each k and n ∈ N, the 2IAP R[n2] (x) satisfy the following set of differential equations: (x)
Ln,k {R[n2] (x)} =
(n + k)!
− k! R[n2] (x),
n!
(2.18)
where (x) Ln,k
:=
k k k! j=1
+
m k k m m
m=1
m
l
l =0
s=0
1
1
s=0
(n − s)!
n − s p
p=1
(n − s)!
n−s−1
n
l
l =0
m k n−1 k m m=1
+
j!
j
xj Djx +
α0(l) β0(m−l)
k k (k − m)!
(j − m)!
j
j =m
−l ) αn(l−) s β0(m−l) + α0(l) βn(m −s
αn(l−) s−p βp(m−l)
k k (k − m)!
(j − m)!
j
j =m
xj−m Djx
x
j−m
Dxn−s+j
.
(2.19)
Proof. Replacing n by n + k in Eq. (2.6), we have R[n2] (x) =
n!
Dkx {Rn+k (x)}, [2]
(n + k)!
which on using Eq. (2.16) on the r.h.s. and applying product rule (2.11) gives Rn (x) =
n!
[2]
m k k m
(n + k)! m=0 m l=0 l m k n −1 k m + m=1
+
m
1
s=0
(n − s)!
α0 β0
k k
j
j =m
1
Dkx−j {xk−m }Djx {R[n2] (x)}
−l) αn(l−) s β0(m−l) + α0(l) βn(m −s
(n − s)! s=0 n −s−1 k n−s k (l) (m−l) k−j k−m n−s+j [2] αn−s−p βp Dx {x }Dx {Rn (x)} .
l =0
n
(l) (m−l)
l
p=1
p
j =m
j
On solving the summation for m = 0 in the first term of the above equation and then using Eq. (2.16) in the resultant equation, we find Rn (x) = [2]
n!
k k
m k k m
k!Rn (x) + {x } {Rn (x)} + (n + k)! j m j=1 m=1 m k k n −1 k k m × Dkx−j {xk−m }Djx {R[n2] (x)} + j =m n
+
[2]
j
Dkx−j
k
m=1
1
s=0
(n − s)!
n−s−1
n − s p=1
p
[2]
Djx
m
αn(l−) s−p βp(m−l)
l
l =0
k k j =m
j
s =0
l=0
l
1
(n − s)!
α0(l) β0(m−l)
−l ) (αn(l−) s β0(m−l) + α0(l) βn(m −s )
Dkx−j
{x
k−m
}
Dxn−s+j
{Rn (x)} , [2]
(2.20)
S. Khan, M. Riyasat / Journal of Computational and Applied Mathematics 306 (2016) 116–132
123
which on making use of Eq. (2.16) again, becomes Rn (x) = [2]
n!
k k k!
m k k m
α0(l) β0(m−l) (n + k)! j ! m l j m=1 l =0 j =1 m k k n −1 1 k (k − m)! j−m j [2] k m × {x }Dx {Rn (x)} + (αn(l−) s j ( j − m )! m l ( n − s )! j =m m=1 s =0 l=0 n −s−1 n k 1 n−s k (k − m)! (m−l) (l) (m−l) (l) × β0 + α0 βn−s ) + αn−s−p βp(m−l) ( n − s )! p j (j − m)! j =m p=1 s =0 k!R[n2] (x) +
xj Djx {Rn[2] (x)} +
× xj−m Dnx −s+j {R[n2] (x)} . Simplifying the above equation, we get assertion (2.18).
Remark 2.1. For k = 1 and k = 2, we deduce the following consequences of Theorem 2.1. Corollary 2.1. For two different sets of Appell polynomials Rn (x) and Rn (x) and with A1 (t ) and A2 (t ) defined by Eqs. (1.11) and (1.12), let A′1 (t ) A1 (t )
=
∞
tn
αn(1)
(2.21a)
n!
n =0
and A′2 (t ) A2 (t )
=
∞
βn(1)
n =0
tn n!
,
(2.21b)
respectively. Then, under assumption (2.15), the following recurrence relation for the 2IAP R[n2] (x) holds true: (1)
(1)
Rn+1 (x) = (x + α0 + β0 )R[n2] (x) + [2]
n −1 n
s
s =0
(αn(1−)s + βn(1−)s )R[s2] (x).
(2.22)
The 1-times shift operators (or simply the shift operators) are given by 1
Θn−(1) := Φn− =
n
Dx
(2.23)
and (1)
(1)
Θn+(1) := (x + α0 + β0 ) +
n −1
1
s =0
(n − s)!
(αn(1−)s + βn(1−)s )Dxn−s .
(2.24)
Corollary 2.2. For two different sets of Appell polynomials Rn (x) and Rn (x) and with A1 (t ) and A2 (t ) defined by Eqs. (1.11) and (1.12), let A′1 (t ) A1 (t )
=
∞
αn(1)
n =0
tn n!
A′′1 (t )
,
A1 (t )
=
∞
αn(2)
n =0
tn
(2.25a)
n!
and A′2 (t ) A2 (t )
=
∞
βn(1)
n =0
tn n!
A′′2 (t )
,
A2 (t )
=
∞
βn(2)
n =0
tn n!
,
(2.25b)
respectively. Then, under assumption (2.15), the following recurrence relation for the 2IAP R[n2] (x) holds true: Rn+2 (x) = [2]
(1)
(1)
(2)
(2)
(1) (1)
x2 + 2(α0 + β0 )x + α0 + β0 + 2α0 β0
+
n −1 n s=0
+2
(1)
(1)
(2)
R[n2] (x) (1)
(1)
(1) (1)
2x(αn−s + βn−s ) + αn−s + βn−s + 2(αn−s β0 + α0 βn−s ) R[s2] (x)
s
n n −s−1 n n−s s=0
(2)
s
p=1
p
αn(1−)s−p βp(1) R[s2] (x).
(2.26)
124
S. Khan, M. Riyasat / Journal of Computational and Applied Mathematics 306 (2016) 116–132
The corresponding shift operators are given by 1
Θn(−2) := Φn−−1 Φn− =
n(n − 1)
D2x
(2.27)
and
(1) (1) (2) (2) (1) (1) Θn(+2) := x2 + 2(α0 + β0 )x + α0 + β0 + 2α0 β0 +
n−1
1
s=0
(n − s)!
+2
(1)
(1)
n−s−1
n
1
s=0
(n − s)!
(2)
(2)
(1)
(1)
(1) (1)
2x(αn−s + βn−s )x + αn−s + βn−s + 2(αn−s β0 + α0 βn−s ) Dnx −s
n − s p
p=1
αn(1−)s−p βp(1) Dxn−s .
(2.28)
Remark 2.2. For k = 1 and k = 2, we deduce the following consequences of Theorem 2.2. Corollary 2.3. For n ∈ N, the 2IAP R[n2] (x) satisfy the following differential equation: (x)
Ln,1 {R[n2] (x)} = n R[n2] (x),
(2.29)
where (x)
(1)
(1)
Ln,1 := (x + α0 + β0 )Dx +
n−1
1
s=0
(n − s)!
(αn(1−)s + βn(1−)s )Dxn−s+1 .
(2.30)
Corollary 2.4. For n ∈ N, the 2IAP R[n2] (x) satisfy the following differential equation: (x)
Ln,2 {R[n2] (x)} = (n2 + 3n) R[n2] (x),
(2.31)
where (x)
(1)
(1)
Ln,2 := 4 x + α0 + β0
+
n −1
1
s=0
(n − s)!
+ 2αn(1−)s β0(1)
+
(1)
(1)
(2)
(2)
(1) (1)
Dx + x2 + 2(α0 + β0 )x + α0 + β0 + 2α0 β0 (1)
(1)
4αn−s + 4βn−s Dnx −s+1 +
(1) (1) 2α0 βn−s
Dnx −s+2
+2
n−1
1
s=0
(n − s)!
n
1
s=0
(n − s)!
(1)
(1)
D2x (2)
(2)
2x(αn−s + βn−s ) + αn−s + βn−s
n−s−1
n − s p=1
p
αn(1−)s−p βp(1) Dxn−s+2 .
(2.32)
In the next section, the integral equations for the Appell polynomials Rn (x) and 2IAP R[n2] (x) are derived. 3. Integral equations Integral equations arise in many scientific and engineering problems. A large class of initial and boundary value problems can be converted to Volterra or Fredholm integral equations. The potential theory contributed more than any field to give rise to integral equations. Mathematical physics models, such as diffraction problems scattering in quantum mechanics, conformal mapping and water waves also contributed to the creation of integral equations. The integral equations for the Appell polynomials Rn (x) and 2-iterated Appell polynomials R[n2] (x) have not been found before. Hence, the integral equations satisfied by these polynomials may be used to express the problems arising in new and emerging areas of sciences. We recall that the Appell polynomials Rn (x) are the solutions of the following differential equation [24, p. 234(2.1)]:
αn−1 (n) αn−2 (n−1) α1 y + y + · · · + y′′ + (x + α0 )y′ − ny = 0. (n − 1)! (n − 2)! 1!
(3.1)
First, we derive the integral equation for the Appell polynomials Rn (x) by making use of the differential equation (3.1) by proving the following result:
S. Khan, M. Riyasat / Journal of Computational and Applied Mathematics 306 (2016) 116–132
125
Theorem 3.1. For the Appell polynomials Rn (x) defined by Eq. (1.1), the following homogeneous Volterra integral equation holds true:
x n −3 φ(x) = − (n − 1)δ2 Pn−2 − (n − 1)(n − 2)δ3 Pn−2 x + Pn−3 − · · · − (n − 1)!δn−1 Pn−2 (n − 3)! n −2 n −4 x x x x xn−3 + Pn−3 + · · · + P2 + P1 − (n − 1)! + δn Pn−2 + Pn−3 (n − 4)! 1! αn−1 (n − 2)! (n − 3)! n−1 n−2 n! x x x x Pn−2 + Pn−3 + · · · + mRm−1 + Rm + · · · + P1 + mRm−1 + 1! αn−1 (n − 1)! (n − 2)! 1! x (n − 1)δ2 + (n − 1)(n − 2)δ3 (x − ξ ) + · · · + (n − 1)!δn−1 (x − ξ )n−3 − 0
x n! + (n − 1)! + δn (x − ξ )n−2 − (x − ξ )n−1 φ(ξ )dξ , αn−1 αn−1
(3.2)
α
where δi := α n−i (i = 2, 3, . . . , n − 1, n) and numerical coefficients αk (k = 1, 2, . . . , n − 1) are given by n−1 A′ (t ) A(t )
=
∞
αn
n =0
tn n!
.
(3.3)
Proof. Consider differential equation (3.1) in the following form: dn y dxn
+ (n − 1)δ2
dn−1 y dxn−1
+ (n − 1)(n − 2)δ3
dn−2 y dxn−2
+ · · · + (n − 1)!δn−1
d2 y dx2
+ (n − 1)!
x dy n! × + δn − y = 0, αn−1 dx αn−1
(3.4)
α
where δi := α n−i (i = 2, 3, . . . , n − 1, n) and αk (k = 1, 2, . . . , n − 1) are the numerical coefficients given by Eq. (3.3). n−1 In view of Eqs. (1.1)–(1.3) the initial conditions are obtained as follows: y(0) = Rm , y′ (0) = mRm−1 , y′′ (0) = m(m − 1)Rm−2 = P1 :=
1 (m − k)Rm−2 , k=0
.. . y(n−2) (0) = m(m − 1) · · · (m − n + 3)Rm−n+2 = Pn−3 :=
n −3
(m − k)Rm−n+2 ,
k=0
y(n−1) (0) = m(m − 1) · · · (m − n + 2)Rm−n+1 = Pn−2 :=
n −2
(m − k)Rm−n+1 ,
k=0
where prime denotes differentiation with respect to x and
Pn−s :=
n −s
(m − k)Rm−n+(s−1) ,
s = n − 1, n − 2, . . . , 3, 2.
k=0
Consider dn y dxn
= φ(x).
Integrating the above equation and using initial conditions (3.5), we have dn−1 y dxn−1 dn−2 y dxn−2
x
φ(ξ )dξ + Pn−2 ,
= 0 x
φ(ξ )dξ 2 + Pn−2 x + Pn−3 ,
= 0
(3.5)
126
S. Khan, M. Riyasat / Journal of Computational and Applied Mathematics 306 (2016) 116–132
.. . x x n −3 x n −4 x n−2 φ(ξ ) d ξ + P = + P + · · · + P2 + P1 , n − 2 n − 3 2 dx (n − 3)! (n − 4)! 1! 0 x n−2 n−3 x x x dy φ(ξ )dξ n−1 + Pn−2 = + Pn−3 + · · · + P1 + mRm−1 , dx (n − 2)! (n − 3)! 1! 0 x xn−1 xn−2 x φ(ξ )dξ n + Pn−2 y= + Pn−3 + · · · + mRm−1 + Rm . (n − 1)! (n − 2)! 1! 0 d2 y
(3.6)
Using expressions (3.6) in Eq. (3.4), we find
x x φ(ξ )dξ 2 + Pn−2 x + Pn−3 φ(ξ )dξ + Pn−2 + (n − 1)(n − 2)δ3 φ(x) + (n − 1)δ2 0 0 x n −3 x xn−4 x φ(ξ )dξ n−2 + Pn−2 + · · · + (n − 1)!δn−1 + Pn−3 + · · · + P2 + P1 (n − 3)! (n − 4)! 1! 0 x x n−2 n−3 x x x + (n − 1)! φ(ξ )dξ n−1 + Pn−2 + δn + Pn−3 + · · · + P1 + mRm−1 αn−1 (n − 2)! (n − 3)! 1! 0 n! x n−1 n−2 x x x − φ(ξ )dξ n + Pn−2 + Pn−3 + · · · + mRm−1 + Rm = 0, αn−1 0 (n − 1)! (n − 2)! 1! which after simplification yields assertion (3.2).
Theorem 3.2. For the 2IAP R[n2] (x) defined by Eq. (1.10), the following homogeneous Volterra integral equation holds true:
ψ(x) = −(n − 1)γ2
n−2
(m − k)T(n−1)−p
− (n − 1)(n − 2)γ3
k=0
+
n−3
(m − k)T(n−1)−p x
k=0
(m − k)T(n−2)−p − · · · − (n − 1)!γn−1
k=0
n−2
n−2 k=0
(m − k)T(n−1)−p
xn−3 + ··· (n − 3)!
2 1 n−2 x x + (m − k)T3−p + (m − k)T2−p − (n − 1)! + γn (m − k) 1! αn−1 + βn−1 k=0 k=0 k=0 1 n −2 x xn−2 n! (m − k)T2−p + ··· + + mT1−p + (m − k) (n − 2)! 1! αn−1 + βn−1 k=0 k=0 xn−1 x x × T(n−1)−p + · · · + mT1−p + T −p − (n − 1)γ2 + (n − 1)(n − 2) (n − 1)! 1! 0 x × γ3 (x − ξ ) + · · · + (n − 1)!γn−1 (x − ξ )n−3 + (n − 1)! + γn (x − ξ )n−2 αn−1 + βn−1 n! − (x − ξ )n−1 ψ(ξ )dξ , αn−1 + βn−1
× T(n−1)−p
α
(3.7)
+β
where γi := α n−i +βn−i (i = 2, 3, . . . , n − 1, n) and the numerical coefficients αk , βk (k = 1, 2, . . . , n − 1) are given by n−1 n−1 A′1 (t ) A1 (t )
=
∞
αn
n =0
tn
(3.8a)
n!
and A′2 (t ) A2 (t ) respectively.
=
∞ n =0
βn
tn n!
,
(3.8b)
S. Khan, M. Riyasat / Journal of Computational and Applied Mathematics 306 (2016) 116–132
127
Proof. Consider differential equation (2.29) (with (2.30)) for the 2IAP R[n2] (x) in the following form: dn−1 y
dn y
d n −2 y
d2 y
+ (n − 1)γ2 n−1 + (n − 1)(n − 2)γ3 n−2 + · · · + (n − 1)!γn−1 2 + (n − 1)! dx dx dx dy x n! × + γn − y = 0, αn−1 + βn−1 dx αn−1 + βn−1
dxn
α
(3.9)
+β
where γi := α n−i +βn−i (i = 2, 3, . . . , n − 1, n) and αk , βk (k = 1, 2, . . . , n − 1) are the numerical coefficients given by Eqs. n−1 n−1 (3.8a) and (3.8b), respectively. In view of Eqs. (1.10)–(1.14), the initial conditions are obtained as follows:
m m
y(0) =
p
p=0
Rm−p Rp = T−p ,
m−1
y′ (0) = m
m − 1 p
p=0
Rm−1−p Rp = mT1−p ,
m−2
y′′ (0) = m(m − 1)
m − 2 p
p=0
Rm−2−p Rp =
1 (m − k)T2−p ,
(3.10)
k=0
.. . y(n−2) (0) = m(m − 1) · · · (m − n + 3)
m−n+2
m − n + 2 p
p=0
y(n−1) (0) = m(m − 1) · · · (m − n + 2)
Rm−n+2−p Rp =
m − n + 1 p
(m − k)T(n−2)−p ,
k=0
m−n+1
p=0
n−3
Rm−n+1−p Rp =
n−2
(m − k)T(n−1)−p ,
k=0
where prime denotes differentiation with respect to x and
Ts−p :=
m−s m−s p
p=0
Rm−s−p Rp ,
s = 0, 1, 2, . . . , n − 2, n − 1.
Consider dn y
= ψ(x), x n −2 ( m − k ) T = ψ(ξ ) d ξ + , ( n − 1 )− p n −1
dxn dn−1 y dx
0
n −2
d
y
=
dxn−2
k=0 x
n −2 n −3 ψ(ξ )dξ 2 + (m − k)T(n−1)−p x + (m − k)T(n−2)−p
0
k=0
.. .
n −2
k =0
2 1 xn−3 x + · · · + ( m − k ) T + ( m − k ) T , 3 − p 2 − p dx2 (n − 3)! 1! 0 k=0 k=0 k=0 x n−2 1 x n −2 x dy = ψ(ξ )dξ n−1 + (m − k)T(n−1)−p + ··· + (m − k)T2−p + mT1−p , dx (n − 2)! 1! 0 k=0 k=0 x n−2 x xn−1 y= ψ(ξ )dξ n + (m − k)T(n−1)−p + · · · + mT1−p + T −p . (3.11) (n − 1)! 1! 0 k=0
d2 y
=
x
ψ(ξ )dξ n−2 +
(m − k)T(n−1)−p
Using expressions (3.11) in Eq. (3.9), we find n −2 x x ψ(x) + (n − 1)γ2 ψ(ξ )dξ + (m − k)T(n−1)−p + (n − 1)(n − 2)γ3 ψ(ξ )dξ 2 0
+
n −2 k =0
0
k=0
(m − k)T(n−1)−p x +
n −3 k=0
(m − k)T(n−2)−p
x + · · · + (n − 1)!γn−1 ψ(ξ )dξ n−2 0
128
S. Khan, M. Riyasat / Journal of Computational and Applied Mathematics 306 (2016) 116–132 n −2
2 1 x n −3 x + ··· + (m − k)T3−p + (m − k)T2−p (n − 3)! 1! k=0 k=0 k=0 n − 2 xn−2 x x + (n − 1)! ψ(ξ )dξ n−1 + (m − k)T(n−1)−p + γn + ··· αn−1 + βn−1 (n − 2)! 0 k =0 n−2 1 x x n! − ψ(ξ )dξ n + (m − k)T(n−1)−p + (m − k)T2−p + mT1−p 1! αn−1 + βn−1 0 k=0 k=0 n −1 x x × = 0, + · · · + mT1−p + T −p (n − 1)! 1!
+
(m − k)T(n−1)−p
which after simplification yields assertion (3.7).
(3.12)
Remark 3.1. We have given an independent proof to derive integral equation (3.2) for the Appell polynomials Rn (x). We note that integral equation (3.2) can also be obtained from integral equation (3.7) of the 2IAP R[n2] (x) by taking p = 0 and α βn−i = 0 (i = 1, 2, 3, . . . , n − 1, n) i.e. replacing γi by α n−i := δi (i = 2, 3, . . . , n − 1, n). n−1
4. Example (λ)
The recurrence relation, shift operators and differential equation for the generalized Bernoulli polynomials Bn (x) and Bernoulli polynomials Bn (x) are derived in [32]. In this section, the recurrence relation, shift operators, differential equation (λ)[2] (x) are derived. and integral equation for the 2IGBP Bn For this, we consider the following example: Example 4.1. Taking A1 (t ) = A2 (t ) =
λ
t et −1
(λ)[2] (that is when the 2IAP Rn[2] (x) reduce to the 2IGBP Bn (x)) and using
generating function (1.4), we find A′1 (t ) A1 (t )
=
A′2 (t ) A2 (t )
= −λ
∞ Bn+1 (1) t n n=0
n + 1 n!
.
(4.1)
Again, in view of Eqs. (2.21a) and (2.21b), we have ∞
αn
n =0
tn n!
=
∞
βn
n =0
tn n!
= −λ
∞ Bn+1 (1) t n n=0
n + 1 n!
,
which on taking n = 0 and using the value B1 (1) =
(4.2) 1 2
gives
λ α0 = β0 = − .
(4.3)
2
Now, substituting the values of the coefficients from Eqs. (4.2) and (4.3) in Eq. (2.22), we find the following recurrence (λ)[2] relation for the 2IGBP Bn (x): (λ)[2]
2] (x) − 2λ Bn+1 (x) = (x − λ)B(λ)[ n
n−1 n Bn−s+1 (1) s=0
s
n−s+1
2] B(λ)[ (x). s
(4.4) (λ)[2]
Similarly, from Eqs. (2.23) and (2.24), we find the following shift operators for the 2IGBP Bn
Θn−(1) := Φn− =
1 n
(x):
Dx
(4.5)
and
Θn+(1) := (x − λ) − 2λ
n−1 Bn−s+1 (1) n−s D . (n − s + 1)! x s=0
(4.6) (λ)[2]
Further, from Eqs. (2.31) and (2.32), the following differential equation for the 2IGBP Bn n −1 Bn−s+1 (1) n−s+1 (x − λ)Dx − 2λ Dx − n Bn(λ)[2] (x) = 0. (n − s + 1)! s =0
(x) is obtained: (4.7)
S. Khan, M. Riyasat / Journal of Computational and Applied Mathematics 306 (2016) 116–132
129
Remark 4.1. Taking λ = 1 in Eqs. (4.5)–(4.7) and in view of relation (1.9), the recurrence relation, shift operators and differential equation for the 2IBP B[n2] (x) are obtained as: Bn+1 (x) = (x − 1)B[n2] (x) − 2 [2]
n −1 n Bn−s+1 (1) s=0
Θn−(1) := Φn− =
1 n
s
n−s+1
B[s2] (x),
(4.8)
Dx ;
(4.9)
n −1 Bn−s+1 (1) n−s D ( n − s + 1)! x s=0
Θn+(1) := (x − 1) − 2
(4.10)
and
(x − 1)Dx − 2
n−1 Bn−s+1 (1) n−s+1 Dx − n B[n2] (x) = 0, (n − s + 1)! s=0
(4.11)
respectively. (λ)
Next, we derive the integral equation for the generalized Bernoulli polynomials Bn (x) from the integral equation of the Appell polynomials Rn (x).
Taking A(t ) =
t et −1
λ
, (that is when the Appell polynomials Rn (x) reduce to the generalized Bernoulli polynomials
(λ)
Bn (x)) and using generating function (1.4), we find A′ (t ) A(t )
= −λ
∞ Bn+1 (1) t n
n + 1 n!
n =0
.
(4.12)
Again, in view of Eq. (2.21a), we have ∞ n =0
αn
tn n!
= −λ
∞ Bn+1 (1) t n
n + 1 n!
n =0
.
(4.13)
Taking n = 0, 1, . . . , n − 2, n − 1 in Eq. (4.13) and equating the coefficients of the same powers of t, we find
α0 = −λB1 (1) = − α1 = −λ
B2 (1) 2
.. . αn−3 = −λ αn−2 = −λ αn−1 = −λ
1
2
2
=−
Bn−2 (1) n−2 Bn−1 (1) n−1 Bn (1) n
λ
since B1 (1) =
λ
12
since B2 (1) =
, 1 6
,
, ,
.
(4.14) α
Using the values from Eq. (4.14), we obtain the coefficients δi := α n−i (i = 2, 3, . . . , n − 1, n). Finally, substituting these n−1 values in integral equation (3.2) of the Appell polynomials Rn (x) and replacing the coefficients Rm of the Appell numbers (λ) by the coefficients Bm of the generalized Bernoulli numbers, the following integral equation for the generalized Bernoulli (λ) polynomials Bn (x) is obtained:
B (1) (1) n! 1 n −2 Pn−2 − n(n − 1) Pn−2 x + Pn−3 − · · · − Bn (1) Bn (1) 12 Bn (1) xn−3 xn−4 x x 1 1 × Pn−2 + Pn−3 + · · · + P2 + P1 + n! − (n − 3)! (n − 4)! 1! λ 2 Bn (1) xn−2 xn−3 x n! n × Pn−2 + Pn−3 + · · · + P1 + mB(λ) m−1 − (n − 2)! (n − 3)! 1! λ Bn (1)
φ(x) = −n
B
n −1
130
S. Khan, M. Riyasat / Journal of Computational and Applied Mathematics 306 (2016) 116–132
xn−1 x n −2 x B (1) x n−1 (λ) n + Pn−3 + · · · + mB(λ) + B − m m−1 (n − 1)! (n − 2)! 1! Bn (1) 0 x B (1) (n − 1)! 1 1 n−2 (x − ξ ) + · · · + (x − ξ )n−3 − (n − 1)! − + (n − 1) Bn (1) 12 Bn (1) λ 2 1 n! 1 × (x − ξ )n−2 + (x − ξ )n−1 φ(ξ )dξ , Bn (1) λ Bn (1) ×
Pn−2
(4.15)
where
Pn−s :=
n −s
(m − k)B(λ) m−n+(s−1) ,
s = n − 1, n − 2, . . . , 3, 2.
k=0
(λ)[2]
Further, we derive the integral equation for the 2-iterated generalized Bernoulli polynomials Bn equation of the 2IAP R[n2] (x).
Taking A1 (t ) = A2 (t ) =
t et −1
λ
(x) from the integral
(λ)[2] , (that is when the 2IAP R[n2] (x) reduce to the 2IGBP Bn (x)) and using generating
function (1.4), we find A′1 (t ) A1 (t )
A′2 (t )
=
A2 (t )
∞ Bn+1 (1) t n
= −λ
n + 1 n!
n=0
.
(4.16)
Again, in view of Eqs. (2.21a) and (2.21b), we have ∞ n =0
αn
tn n!
=
∞
βn
n =0
tn
= −λ
n!
∞ Bn+1 (1) t n n=0
n + 1 n!
.
(4.17)
Taking n = 0, 1, . . . , n − 2, n − 1 in Eq. (4.17) and equating the coefficients of the same powers of t, we find
α0 = β0 = −λB1 (1) = − α1 = β1 = −λ
B2 (1) 2
.. . αn−3 = βn−3 = −λ αn−2 = βn−2 = −λ αn−1 = βn−1 = −λ
λ 2
=−
λ
12
Bn−2 (1) n−2 Bn−1 (1) n−1 Bn (1) n
since B1 (1) =
1
, 1 since B2 (1) = , 2
6
, ,
.
(4.18) α
+β
Making use of the values from Eq. (4.18), we obtain the coefficients γi := α n−i +βn−i (i = 2, 3, . . . , n − 1, n). Finally, n−1 n−1 substituting these values in integral equation (3.7) of the 2IAP R[n2] (x) and replacing the Appell coefficients Rm , Rp by the (λ)
(λ)
coefficients Bm , Bp
of the generalized Bernoulli numbers, the following integral equation for the 2-iterated generalized (λ)[2]
(x) is obtained: n −2 n −2 B (1) B (1) n −2 n −1 ψ(x) = −n (m − k)T(n−1)−p − n(n − 1) (m − k)T(n−1)−p x Bn (1) Bn (1) k=0 k=0
Bernoulli polynomials Bn
+
n −3 n −2 x n −3 n! 1 (m − k)T(n−2)−p − · · · − (m − k)T(n−1)−p + ··· 12 Bn (1) (n − 3)! k=0 k=0
+
2 1 n−2 x x 1 1 (m − k)T3−p + (m − k)T2−p + n! − (m − k)T(n−1)−p 1! 2λ 2 Bn (1) k=0 k=0 k=0
1 n −2 x n! n + mT1−p − (m − k)T2−p (m − k)T(n−1)−p (n − 2)! 1! 2λ Bn (1) k=0 k=0 x B (1) x xn−1 Bn−1 (1) n −2 × + · · · + mT1−p + T−p − n + (n − 1) (x − ξ ) + · · · (n − 1)! 1! Bn (1) Bn (1) 0
×
xn−2
+ ··· +
S. Khan, M. Riyasat / Journal of Computational and Applied Mathematics 306 (2016) 116–132
131
x n! 1 (n − 1)! 1 1 1 + (x − ξ )n−3 − (n − 1)! (x − ξ )n−2 + − 12 Bn (1) 2λ 2 Bn (1) 2λ Bn (1) n −1 ψ(ξ )dξ , × (x − ξ )
(4.19)
where
Ts−p :=
m−s m−s p=0
p
(λ)
Bm−s−p B(λ) p ,
s = 0, 1, 2, . . . , n − 2, n − 1.
(λ)
Remark 4.2. By taking λ = 1 in integral equations (4.15) and (4.19) of the generalized Bernoulli polynomials Bn (x) and (λ)[2] 2IGBP Bn (x), the integral equations of the Bernoulli polynomials Bn (x) and 2IBP B[n2] (x) can be derived. 5. Concluding remarks The orthogonal polynomials in general and the classical orthogonal polynomials in particular have been the object of extensive works. They are connected with numerous problems of applied mathematics, theoretical physics, chemistry, approximation theory and several other mathematical branches. In particular, their applications are being widely used in theories as Padé approximants, continued fractions, spectral study of Schrödinger discrete operators, polynomial solutions of second-order differential equations and others. The notions of d-dimensional orthogonality for polynomials [33], vectorial orthogonality [34] or simultaneous orthogonality [35] are the generalizations of ordinary orthogonality for polynomials. Such polynomials are characterized by the fact that they satisfy a d + 1-order recurrence relationship, that is a relation between d + 2 consecutive polynomials [34]. All these new notions of d-orthogonality for polynomials and, equivalently, 1/d-orthogonality [36] appear as particular cases of the general notion of biorthogonality studied in [37]. Recently they have been the subject of numerous investigations and applications. In particular, they are connected with the study of vector Padé approximants [34,38], simultaneous Padé approximants [35] and other problems such as vectorial continued fractions, polynomials solutions of the higher-order differential equations, spectral study of multi diagonal non-symmetric operators [39]. In this article, a set of finite order recurrence relations and differentials equations for the Appell and 2-iterated Appell polynomials Rn (x) and R[n2] (x), respectively are established. The integral equations for the Appell and 2-iterated Appell polynomials Rn (x) and R[n2] (x), respectively are also derived. Since, the Appell and 2-iterated Appell polynomials are important from the point of view of their applications in various fields, therefore the differential and integral equations satisfied by these polynomials may be used to solve the existing as well as new emerging problems in certain branches of science. The problems of finding all polynomial sequences, which are at the same time Appell polynomials and d-orthogonal is considered in [23]. The recurrence relation and differential equations of the 2IAP A[n2] (x) established in this article may be used to study the d-orthogonality of these polynomials. This aspect may be considered in further investigation. Acknowledgments The authors are thankful to the reviewer for several useful comments and suggestions towards the improvement of this paper. This work has been done under Senior Research Fellowship (Award letter No. F1-17.1/2012–13, MANF-MUS-UTT-9243) awarded to the second author by the University Grants Commission, Government of India, New Delhi. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
P. Appell, Sur une classe de polynômes, Ann. Sci. École Norm. Sup. 9 (2) (1880) 119–144. S. Roman, The Umbral Calculus, Academic Press, New York, 1984. H.T. Davis, The Theory of Linear Operator, Principia Press, Bloomington, Indiana, 1936. A. Di Bucchianico, D. Loeb, A selected survey of umbral calculus, Electron. J. Combin. DS3 (2000) 1–34. A. Erdélyi, W. Magnus, F. Oberhettinger, F.G. Tricomi, Higher Transcendental Functions, Vol. I, McGraw-Hill Book Company, New York, Toronto, London, 1953. A. Erdélyi, W. Magnus, F. Oberhettinger, F.G. Tricomi, Higher Transcendental Functions, Vol. III, McGraw-Hill Book Company, New York, Toronto, London, 1955. E.D. Rainville, Special Functions, Reprint of 1960 First Edition, Chelsea Publishing Company, Bronx, New York, 1971. J. Stoer, Introduzione all’Analisi Numerica, Zanichelli, Bologna, 1972. Subuhi Khan, N. Raza, 2-Iterated Appell polynomials and related numbers, Appl. Math. Comput. 219 (17) (2013) 9469–9483. Subuhi Khan, M. Riyasat, A study on the 2-Iterated q-Appell polynomials from determinantal point of view, J. Math. Anal. Appl. (2016) submitted for publication. T.H. Cormen, C.E. Leiserson, R.L. Rivest, C. Stein, Introduction to Algorithms, MIT Press, 2009. R. Sedgewick, F. Flajolet, An Introduction to the Analysis of Algorithms, Addison-Wesley, 2013. J. Thomas Sargent, Dynamic Macroeconomic Theory, Harvard University Press, 1987.
132 [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39]
S. Khan, M. Riyasat / Journal of Computational and Applied Mathematics 306 (2016) 116–132 W. Johnson, A Treatise on Ordinary and Partial Differential Equations, John Wiley and Sons, 1913, in University of Michigan Historical Math Collection. R.P. Boas, R.C. Buck, Polynomial Expansions of Analytic Functions, Springer-Verlag, Berlin, Gottingen, Heidelberg, 1958. I.M. Sheffer, A differential equation for Appell polynomials, Bull. Amer. Math. Soc. 41 (12) (1935) 914–923. J. Shohat, The relation of the classical orthogonal polynomials to the polynomials of Appell, Amer. J. Math. 58 (3) (1936) 453–464. G. Bretti, C. Cesarano, P.E. Recci, Laguerre-type exponential and generalized Appell polynomials, Comput. Math. Appl. 48 (5–6) (2004) 833–839. G. Bretti, M.X. He, P.E. Ricci, On quadrature rules associated with Appell polynomials, Int. J. Appl. Math. 11 (2002) 1–14. G. Bretti, P. Natalini, P.E. Ricci, Generalizations of the Bernoulli and Appell polynomials, Abstr. Appl. Anal. 7 (2004) 613–623. G. Bretti, P.E. Ricci, Multidimensional extensions of the Bernoulli and Appell polynomials, Taiwanese J. Math. 8 (2004) 415–428. G. Dattoli, P.E. Ricci, C. Cesarano, Differential equations for Appell type polynomials, Fract. Calc. Appl. Anal. 5 (2002) 69–75. K. Douak, The relation of the d-orthogonal polynomials to the Appell polynomials, J. Comput. Appl. Math. 70 (2) (1996) 279–295. M.X. He, P.E. Ricci, Differential equation of Appell polynomials via the factorization method, J. Comput. Appl. Math. 139 (2002) 231–237. M.E.H. Ismail, Remarks on differential equation of Appell polynomials via the factorization method, J. Comput. Appl. Math. 154 (2003) 243–245. M.A. Özarslan, B. Yilmaz, A set of finite order differential equations for the Appell polynomials, J. Comput. Appl. Math. 259 (2014) 108–116. 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. J.F. Steffensen, The poweriod, an extension of the mathematical notion of power, Acta Math. 73 (1941) 333–366. G. Dattoli, C. Cesarano, D. Sacchetti, A note on the monomiality principle and generalized polynomials, J. Math. Anal. Appl. 227 (1997) 98–111. L. Infeld, T.E. Hull, The factorization method, Rev. Modern Phys. 23 (1951) 21–68. A.J. Jerry, Introduction to Integral Equations with Applications (Second Edition), John Wiley and Sons, Inc., New York, Chichester, Weinheins, Brisbane, Singapore, Toronto, 1999. Da-Qian Lu, Some properties of Bernoulli polynomials and their generalizations, Appl. Math. Lett. 24 (2011) 746–751. P. Maroni, L’orthogonalité et les récurrences de polynoˆ mes d’ordre supérieur á deux, Ann. Fac. Sci. Toulouse 10 (1989) 105–139. J. Van Iseghem, Approximants de padé vectoriels (Thése d’état), Univ. des Sciences et Techniques de Lille-Flandre-Artois, 1987. M.G. de Bruin, Simultaneous padé approximation and orthogonality, in: Lectures Notes in Math., Vol. 1171, Springer, Berlin, 1985, pp. 74–83. A. Boukhemis, P. Maroni, Une caractérisation des polynoˆ mes strictement 1/p-orthogonaux de type Sheffer, étude du cas p = 2, J. Approx. Theory 54 (1988) 67–91. C. Brezinski, Biorthogonality and its Applications to Numerical Analysis, Marcel Dekker, New York, 1992. J. Van Iseghem, Vector orthogonal relations, vector QD-algorithm, J. Comput. Appl. Math. 19 (1987) 141–150. A.I. Aptekarev, V.A. Kaliaguine, W. Van Assche, Criterion for the resolvent set of non-symmetric tridiagonal operators, Proc. Amer. Math. Soc. 123 (8) (1995) 2423–2430.
Contents lists available at ScienceDirect
Journal of Computational and Applied Mathematics journal homepage: www.elsevier.com/locate/cam
Differential and integral equations for the 2-iterated Appell polynomials Subuhi Khan ∗ , Mumtaz Riyasat Department of Mathematics, Aligarh Muslim University, Aligarh, India
article
abstract
info
Article history: Received 14 March 2015 Received in revised form 29 February 2016 MSC: 33E20 33E30 Keywords: Differential equations 2-iterated Appell polynomials Integral equations
In this article, a set of differential equations of finite order (kth order, k ∈ N) for the 2-iterated Appell polynomials are derived. Particular cases k = 1 and k = 2 are also considered. The integral equations for the Appell and 2-iterated Appell polynomials are established. Further, as an illustration the differential and integral equations for the 2-iterated generalized Bernoulli polynomials are derived. This article is first attempt in the direction of deriving integral equations for the Appell and 2-iterated Appell families and for some members belonging to these families. © 2016 Elsevier B.V. All rights reserved.
1. Introduction and preliminaries One of the important classes of polynomial sequences is the class of Appell polynomial sequences [1]. The Appell polynomial sequences appear in different applications in pure and applied mathematics. These 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 sequences. Properties of the Appell sequences are naturally handled within the framework of modern classical umbral calculus by Roman [2]. In 1936 an initial bibliography was provided by Davis [3]. Di Bucchianico [4] summarized and documented more than five hundred old and new findings related to the study of Appell sequences. The Appell polynomials Rn (x) [1] are defined by the following generating function: A(t )ext =
∞
Rn (x)
n =0
tn n!
,
(1.1)
where A(t ) has (at least the formal) expansion: A(t ) =
∞ n =0
Rn
tn n!
,
R0 := A(0) ̸= 0;
Rn := Rn (0).
(1.2)
It is easy to see that for any A(t ) the derivative of Rn (x) satisfies R′n (x) = n Rn−1 (x).
∗
Corresponding author. E-mail addresses: [email protected] (S. Khan), [email protected] (M. Riyasat).
http://dx.doi.org/10.1016/j.cam.2016.03.039 0377-0427/© 2016 Elsevier B.V. All rights reserved.
(1.3)
S. Khan, M. Riyasat / Journal of Computational and Applied Mathematics 306 (2016) 116–132
117
(α)
(α)
The Bernoulli and Euler polynomials Bn (x) and En (x) together with their familiar generalizations Bn (x) and En (x), respectively of (real or complex) order α [5,6] belong to the class of Appell sequences. t For A(t ) = et − , Eq. (1.1) gives the generating function for the Bernoulli polynomials [5]: 1 t
et − 1
∞
ext =
Bn (x)
n=0
tn n!
,
|t | < 2π .
(1.4)
The Bernoulli polynomials Bn (x) play an important role in various expansions and approximation formulas, which are useful both in analytic theory of numbers and in classical and numerical analysis. These polynomials provide solutions to various problems of engineering and physics. The Bernoulli polynomials are employed in the integral representation of differentiable periodic functions and play an important role in the approximation of such functions by means of polynomials. The Bernoulli numbers Bn := Bn (0) are defined by the following generating function [5]: t
et − 1
=
∞
Bn
n =0
tn n!
.
(1.5)
Moreover, we have 1
Bn (0) = (−1)n Bn (1) =
1
, n ∈ N0 . (1.6) 21−n − 1 2 The Bernoulli numbers Bn enter in many mathematical formulas, such as the Taylor expansion in a neighborhood of the origin of the trigonometric and hyperbolic tangent and cotangent functions [7], the sums of powers of natural numbers and residual term of the Euler Maclaurin quadrature formula [8]. Thus, the Bernoulli polynomials and Bernoulli numbers have deep connections with number theory and occur in combinatorics. (λ) Further, we recall that the generalized Bernoulli polynomials Bn (x) defined by the generating function [6]: t
et
λ
−1
ext =
∞
B(λ) n (x)
n =0
tn n!
Bn
,
|t | < 2π ,
belong to the class of Appell sequences with A(t ) =
(1.7)
t et −1
λ
(λ)
and generalized Bernoulli numbers Bn
:= B(λ) n (0) are defined
by the following generating function: t
et
λ
−1
=
∞
B(λ) n
n =0
tn n!
.
(1.8)
From Eqs. (1.4) and (1.7), it follows that B(n1) (x) = Bn (x).
(1.9)
Recently, Subuhi Khan and N. Raza [9] introduced the 2-iterated Appell polynomials by combining two different sets of Appell polynomials. We recall that the 2-iterated Appell polynomials (2IAP) R[n2] (x) are defined by the generating function [9, p. 9471(2.5)]: A1 (t ) A2 (t ) ext =
∞ n =0
R[n2] (x)
tn n!
,
(1.10)
where A1 (t ) =
∞
Rn
n =0
A2 (t ) =
∞
tn n!
Rn
n =0
,
tn n!
,
A1 (0) ̸= 0; A2 (0) ̸= 0;
Rn := Rn (0),
Rn := Rn (0).
(1.11)
(1.12)
The series definition for the 2IAP R[n2] (x) is given as [9, p. 9471(2.9)]: R[n2] (x) =
n n m=0
m
Rm Rn−m (x),
(1.13)
Rm Rn−m ,
(1.14)
where R[n2] (0) =
n n m=0
m
denotes the 2-iterated Appell numbers.
118
S. Khan, M. Riyasat / Journal of Computational and Applied Mathematics 306 (2016) 116–132
We recall that, if pn (x) and qn (x) = pn (x) is defined to be the sequence [2] qn (p(x)) =
n
n
k =0
qn,k xk are sequences of polynomials, then the umbral composition of qn (x) with
qn,k pk (x),
(1.15)
k =0
which is equivalent to condition (1.13). The set of all Appell sequences is closed under the operation of umbral compositions of polynomial sequences. Under this operation the set of all Appell sequences is an abelian group. Since the generating function of the 2IAP is of the form A⋆ (t )ext , with A⋆ (t ) as the product of two similar functions of t. Therefore, the set of all 2IAP sequences also form an abelian group under the operation of umbral composition. With the help of determinantal form of the 2IAP considered in [10], it may be possible to compute the coefficients or the value in a chosen point, for particular sequences of the 2IAP family, through an efficient and stable Gaussian algorithm. It can be useful in finding the solution of general linear interpolation problem. It has been shown in [9], that corresponding to each member belonging to the Appell family, there exists a new mixed special polynomial belonging to the 2IAP family. We recall that corresponding to the Bn (x) defined by Eq. (1.4), the 2-iterated Bernoulli polynomials (2IBP) B[n2] (x) are defined by the generating function [9, p. 9474(Table 2.1)]: t
et
−1
2
ext =
∞
B[n2] (x)
n =0
tn n!
.
(1.16) (λ)
Similarly, corresponding to the Bn (x) defined by Eq. (1.7), the 2-iterated generalized Bernoulli polynomials (2IGBP) (x) are defined by the generating function [9]: Bn (λ)[2]
t
2λ
et − 1
ext =
∞
2] B(λ)[ (x) n
n =0
tn n!
.
(1.17)
The recurrence relations are of fundamental importance in analysis of algorithms [11,12]. In digital signal processing, recurrence relations can model feedback in a system, where outputs at one time become inputs for future time. They thus arise in infinite impulse response (IIR) digital filters. The linear recurrence relations are also used extensively in both theoretical and empirical economics [13]. The study of differential equations is a wide field in pure and applied mathematics, physics, and engineering. The problems arising in different areas of science and engineering are usually expressed in terms of differential equations, which in most of the cases have special functions as their solutions. During the past three decades, the development of nonlinear analysis, dynamical systems and their applications to science and engineering has stimulated renewed enthusiasm for the theory of differential equations. Differential equations play an important role in modeling virtually every physical, technical, or biological process, from celestial motion, to bridge design, to interactions between neurons. Many fundamental laws of physics and chemistry can be formulated as differential equations. In biology and economics, differential equations are used to model the behavior of complex systems [14]. Sequences of Appell polynomials have been well studied because of their remarkable applications in mathematical and numerical analysis, as well as in number theory, as both classic literature [1,15,2,16,17] and more recent literature [18–25] testify. One aspect of such study for these sequences is to find differential equation and recurrence relations for Appell sequences. We recall some preliminaries from [24,26]. Let {pn (x)}∞ n=0 be a sequence of polynomials such that deg(pn (x)) = n, (n ∈ N0 := {0, 1, 2, . . .}). The differential operators Θn− and Θn+ satisfying the properties
Θn− {pn (x)} = pn−1 (x),
(1.18a)
Θn {pn (x)} = pn+1 (x),
(1.18b)
+
are called derivative and multiplicative operators, respectively. The monomiality principle [27,28] and the associated operational rules are used in [29] to explore new classes of isospectral problems leading to nontrivial generalizations of special functions. The polynomial sequence {pn (x)}∞ n=0 is called quasi-monomial, if and only if the above conditions are satisfied. Most of the properties of the families of polynomials associated with these operators can be deduced using operator rules with the Θn− and Θn+ operators. Obtaining the derivative and multiplicative operators of a given family of polynomials gives rise to some useful properties such as
(Θn−+1 Θn+ ){pn (x)} = pn (x),
(1.19a)
(Θn+−1 Θn+−2
(1.19b)
. . . Θ2 Θ1 Θ0 ){p0 (x)} = pn (x). +
+
+
Note that, if Θn− and Θn+ are differential realizations, then the above equations give the differential equation satisfied by pn (x). The technique used in obtaining differential equations via (1.19a) and (1.19b) is known as the factorization method.
S. Khan, M. Riyasat / Journal of Computational and Applied Mathematics 306 (2016) 116–132
119
The main idea of the factorization method is to find the derivative operator Θn− and multiplicative operator Θn+ such that Eq. (1.19a) holds. He and Ricci [24] used the classical factorization method introduced by Infeld and Hull [30] to study finite order differential equation for the Appell polynomials Rn (x). Motivated by this approach Özarslan and Yilmaz [26] extended this method to obtain a set of finite order (kth order, −(k) +(k) k ∈ N) differential equations for Rn (x) by constructing the operators Θn and Θn satisfying
Θn−(k) {pn (x)} = pn−k (x), +(k)
Θn
(1.20a)
{pn (x)} = pn+k (x), −(k)
(1.20b)
+(k)
the operators Θn and Θn are called k-times derivative and k-times multiplicative operators, respectively. Obtaining these operators for a given polynomial set will provide several useful relations for the corresponding polynomial set. Such −(k) +(k) as, when Θn and Θn are differential operators then, for each k ∈ N, the relation k) +(k) (Θn−( ){pn (x)} = pn (x), +k Θn
(1.21)
gives us the differential equations for this polynomial set. This method is called generalized factorization method [26] and it can be used to obtain a set of differential equations for pn (x), because for each k ∈ N, we have one differential equation for this polynomial. On the other hand, if n = mk + r then, by using few number of operators, Eq. (1.19b) can be expressed as +(k) +(k) +(k) + (Θn+−1 . . . Θmk Θ(m−1)k . . . Θk Θ0 ){p0 (x)} = pn (x).
(1.22)
Some problems which have their mathematical representation appear directly and in a natural way in terms of integral equations. Other problems, whose direct representations are in terms of differential equations and their auxiliary conditions may also be reduced to integral equations. Integral equations are important in many applications [31]. Problems in which integral equations encountered include radiative energy transfer and oscillations of a string, membrane or axle. Oscillation problems may also be solved as differential equations. In this article, recurrence relations and differential equations for the 2-iterated Appell polynomials are derived. This article is first attempt in the direction of introducing the integral equations for the Appell and 2-iterated Appell polynomials. +(k) −(k) (k ∈ N) are determined and a set of finite order differential equations for and Θn In Section 2, the shift operators Θn [2] the 2IAP Rn (x) is derived. The differential equations for k = 1 and k = 2 are also obtained as special cases. In Section 3, the integral equations for the Appell polynomials Rn (x) and 2IAP Rn[2] (x) are established. Finally, in Section 4, the shift operators, (λ)[2]
differential equations and integral equations for the 2IGBP Bn
(x) are derived.
2. Recurrence relations and differential equations −(k)
In this section, a set of finite order recurrence relation for the 2IAP R[n2] (x) is established and the shift operators Θn
and
+(k)
, for each k ∈ N are determined. Further, a set of finite order differential equations for the 2IAP R[n2] (x) is derived. In order to establish the recurrence relations, we prove the following result:
Θn
Theorem 2.1. For two different sets of Appell polynomials Rn (x) and Rn (x) and with A1 (t ) and A2 (t ) defined by Eqs. (1.11) and (1.12), let (m)
A1 (t ) A1 (t )
=
∞
αn(m)
n =0
tn
(2.1a)
n!
and (m)
A2 (t ) A2 (t )
=
∞
βn(m)
n =0
tn n!
,
(2.1b)
respectively. Then, the following finite set of recurrence relations for the 2IAP R[n2] (x) holds true: [2] Rn+k (x)
=
k k m=0
m
k−m
x
m m l=0
l
(l) (m−l) [2]
α0 β0
−l ) (l) (m−l) × (α0(l) βn(m )+ −s + αn−s β0
Rn (x) +
k k m=1
m
n n −s−1 n n−s s =0
s
p=1
p
x
k−m
m n −1 m n l =0
l
s=0
s
αn(l−) s−p βp(m−l) R[s2] (x).
(2.2)
120
S. Khan, M. Riyasat / Journal of Computational and Applied Mathematics 306 (2016) 116–132
Moreover, the k-times shift operators are given by n
Θn−(k) :=
Φm− =
m=n−k+1
n
1
m m=n−k+1
Dx =
(n − k)! k Dx n!
(2.3)
and +(k)
Θn
:=
k k m
m=0
x
k−m
m m l =0
−l ) × (α0(l) βn(m −s
+
l
(l) (m−l)
α0 β0
+
k k m=1
αn(l−) s β0(m−l) )Dnx −s
n
+
m
1
s =0
(n − s)!
x
k−m
m n −1 m l
l =0 n−s−1
n − s p
p=1
s=0
1
(n − s)!
αn(l−) s−p βp(m−l) Dnx −s
,
(2.4)
where Dx := ∂∂x . Proof. Let ∞
G(x, t ) = A1 (t )A2 (t )ext :=
R[n2] (x)
n=0
tn n!
.
(2.5)
Differentiating Eq. (2.5) k-times with respect to x and using the fact that
∂ kG = t k G(x, t ), ∂ xk we find ∞
R[n2] (x)
t n +k
n =0
n!
=
∞
Dkx {R[n2] (x)}
n=0
tn n!
,
which on equating the coefficients of same powers of t gives
(R[n2] )(k) (x) =
n!
Rn−k (x). [2]
(n − k)!
Since the operator Φn− =
1 D n x
(2.6)
satisfies the following operational relation [9]:
Φn− {R[n2] (x)} = R[n2−] 1 (x).
(2.7)
Therefore, considering the derivative operator as:
Φn− =
1 n
Dx
(2.8)
and in view of Eq. (2.6), we have
Θn−(k) {R[n2] (x)} :=
n
Φm− {R[n2] (x)} = R[n2−] k (x).
(2.9)
m=n−k+1
Using Eqs. (2.6) and (2.8) on the r.h.s. of Eq. (2.9), we find
Θn−(k) {R[n2] (x)} :=
n m=n−k+1
Φm− {R[n2] (x)} =
n
1
m m=n−k+1
Dx {R[n2] (x)} =
(n − k)! k [2] Dx {Rn (x)}, n!
(2.10)
which proves that the k-times derivative operator is given by (2.3). We differentiate the relation G(x, t ) = A1 (t )A2 (t )ext by using formula:
k ∂k k k−m { f ( t ) g ( t )} = Dm {g (t )}, t {f (t )}Dt ∂tk m m=0 so that we have
k ∂ k G(x, t ) k k−m xt = Dm {e } t {A1 (t )A2 (t )}Dt ∂tk m m=0
(2.11)
S. Khan, M. Riyasat / Journal of Computational and Applied Mathematics 306 (2016) 116–132
121
that is
k m ∂ k G(x, t ) k m l xt k−m = e x Dt {A1 (t )}Dtm−l {A2 (t )}, k ∂t m l m=0 l =0 which on multiplying and dividing by A1 (t )A2 (t ) gives
(l) (m−l) k m ∂ k G(x, t ) k m A1 (t ) A2 (t ) k−m = G ( x , t ) x . k ∂t m l A1 (t ) A2 (t ) m=0 l=0
(2.12)
Now, differentiating Eq. (2.5) k-times with respect to t and then using expression (2.12) in the resultant equation, we have G(x, t )
k k m
m=0
(l) (m−l) m m A (t ) A (t ) 1
xk−m
A1 (t )
l
l =0
2
=
A2 (t )
∞
Rn+k (x) [2]
n =0
tn n!
,
which on making use of Eqs. (2.5), (2.1a) and (2.1b) on the l.h.s. becomes ∞
R s ( x) [2]
s =0
k ts
s! m=0
m
k
x
k−m
m ∞ n m n l
l =0
p
n=0 p=0
t αn(l−) p βp(m−l)
n
n!
=
∞
Rn+k (x) [2]
n =0
tn n!
.
(2.13)
Regulating the series in Eq. (2.13) and then equating the coefficients of the same powers of t, we find [2] Rn+k (x)
=
k k m
m=0
x
k−m
m n n −s m n n − s l
l =0
s
s=0
p
p=0
αn(l−) s−p βp(m−l) Rs[2] (x),
or, equivalently Rn+k (x) = [2]
k k m
m=0
+
xk−m
l
l =0
k k
m=0
m n n−s−1 m n n − s
m
xk−m
s
s=0
p=0
m n m n
l
l =0
s=0
p
s
αn(l−) s−p βp(m−l) R[s2] (x)
−l) [2] α0(l) βn(m −s Rs (x).
(2.14)
Since 1, 0,
αn(0) = βn(0) = δn,0 :=
n=0 otherwise.
(2.15)
Therefore, we have [2] Rn+k (x)
=
k k m=0
×
m
k −m
x
m m l =0
n−1 n
s
s=0
l
(l) (m−l) [2]
α0 β0
Rn (x) +
k k m=1
−l) [2] α0(l) βn(m −s Rs (x)
+
m
xk−m
l =0
n n −s−1 n n−s s=0
s
p=0
m m
p
l
αn(l−) s−p βp(m−l) R[s2] (x)
or, equivalently [2] Rn+k (x)
=
k k m=0
×
m
x
n−1 n
s
n n s=0
m m l =0
s=0
+
k −m
s
l
(l) (m−l) [2]
α0 β0
Rn (x) +
k k m=1
−l) [2] α0(l) βn(m −s Rs (x) +
m
xk−m
n n −s−1 n n−s s=0
s
p=1
p
m m l =0
l
αn(l−) s−p βp(m−l) R[s2] (x)
αn(l−) s β0(m−l) R[s2] (x) ,
which on rearranging the summations in the second term of the r.h.s. yields assertion (2.2).
,
122
S. Khan, M. Riyasat / Journal of Computational and Applied Mathematics 306 (2016) 116–132
Further, in view of Eqs. (2.6) and (2.9), we have n
R[s2] (x) =
Φm− {R[n2] (x)} =
m=s+1
s! n!
Dnx −s [R[n2] (x)].
(2.16)
Using Eq. (2.16) in recurrence relation (2.2), we get
[2] Rn+k (x)
=
k k m
m=0
xk−m
m m
l
l =0
−l ) × (α0(l) βn(m −s
+
α0(l) β0(m−l) +
k k m
m=1
αn(l−) s β0(m−l) )Dnx −s
+
1
s=0
(n − s)! +(k)
m n −1 m
l
l=0
s=0
n−s−1
n
which proves that the k-times multiplicative operator Θn
xk−m
n − s
(n − s)!
αn(l−) s−p βp(m−l) , Dxn−s
p
p=1
1
is given by Eq. (2.4).
R[n2] (x)
(2.17)
Deriving a set finite order differential equations for the 2IAP Rn (x) is a new investigation. A set of finite order differential equations for the 2IAP R[n2] (x) for each k ∈ N is derived by proving the following result: [2]
Theorem 2.2. For each k and n ∈ N, the 2IAP R[n2] (x) satisfy the following set of differential equations: (x)
Ln,k {R[n2] (x)} =
(n + k)!
− k! R[n2] (x),
n!
(2.18)
where (x) Ln,k
:=
k k k! j=1
+
m k k m m
m=1
m
l
l =0
s=0
1
1
s=0
(n − s)!
n − s p
p=1
(n − s)!
n−s−1
n
l
l =0
m k n−1 k m m=1
+
j!
j
xj Djx +
α0(l) β0(m−l)
k k (k − m)!
(j − m)!
j
j =m
−l ) αn(l−) s β0(m−l) + α0(l) βn(m −s
αn(l−) s−p βp(m−l)
k k (k − m)!
(j − m)!
j
j =m
xj−m Djx
x
j−m
Dxn−s+j
.
(2.19)
Proof. Replacing n by n + k in Eq. (2.6), we have R[n2] (x) =
n!
Dkx {Rn+k (x)}, [2]
(n + k)!
which on using Eq. (2.16) on the r.h.s. and applying product rule (2.11) gives Rn (x) =
n!
[2]
m k k m
(n + k)! m=0 m l=0 l m k n −1 k m + m=1
+
m
1
s=0
(n − s)!
α0 β0
k k
j
j =m
1
Dkx−j {xk−m }Djx {R[n2] (x)}
−l) αn(l−) s β0(m−l) + α0(l) βn(m −s
(n − s)! s=0 n −s−1 k n−s k (l) (m−l) k−j k−m n−s+j [2] αn−s−p βp Dx {x }Dx {Rn (x)} .
l =0
n
(l) (m−l)
l
p=1
p
j =m
j
On solving the summation for m = 0 in the first term of the above equation and then using Eq. (2.16) in the resultant equation, we find Rn (x) = [2]
n!
k k
m k k m
k!Rn (x) + {x } {Rn (x)} + (n + k)! j m j=1 m=1 m k k n −1 k k m × Dkx−j {xk−m }Djx {R[n2] (x)} + j =m n
+
[2]
j
Dkx−j
k
m=1
1
s=0
(n − s)!
n−s−1
n − s p=1
p
[2]
Djx
m
αn(l−) s−p βp(m−l)
l
l =0
k k j =m
j
s =0
l=0
l
1
(n − s)!
α0(l) β0(m−l)
−l ) (αn(l−) s β0(m−l) + α0(l) βn(m −s )
Dkx−j
{x
k−m
}
Dxn−s+j
{Rn (x)} , [2]
(2.20)
S. Khan, M. Riyasat / Journal of Computational and Applied Mathematics 306 (2016) 116–132
123
which on making use of Eq. (2.16) again, becomes Rn (x) = [2]
n!
k k k!
m k k m
α0(l) β0(m−l) (n + k)! j ! m l j m=1 l =0 j =1 m k k n −1 1 k (k − m)! j−m j [2] k m × {x }Dx {Rn (x)} + (αn(l−) s j ( j − m )! m l ( n − s )! j =m m=1 s =0 l=0 n −s−1 n k 1 n−s k (k − m)! (m−l) (l) (m−l) (l) × β0 + α0 βn−s ) + αn−s−p βp(m−l) ( n − s )! p j (j − m)! j =m p=1 s =0 k!R[n2] (x) +
xj Djx {Rn[2] (x)} +
× xj−m Dnx −s+j {R[n2] (x)} . Simplifying the above equation, we get assertion (2.18).
Remark 2.1. For k = 1 and k = 2, we deduce the following consequences of Theorem 2.1. Corollary 2.1. For two different sets of Appell polynomials Rn (x) and Rn (x) and with A1 (t ) and A2 (t ) defined by Eqs. (1.11) and (1.12), let A′1 (t ) A1 (t )
=
∞
tn
αn(1)
(2.21a)
n!
n =0
and A′2 (t ) A2 (t )
=
∞
βn(1)
n =0
tn n!
,
(2.21b)
respectively. Then, under assumption (2.15), the following recurrence relation for the 2IAP R[n2] (x) holds true: (1)
(1)
Rn+1 (x) = (x + α0 + β0 )R[n2] (x) + [2]
n −1 n
s
s =0
(αn(1−)s + βn(1−)s )R[s2] (x).
(2.22)
The 1-times shift operators (or simply the shift operators) are given by 1
Θn−(1) := Φn− =
n
Dx
(2.23)
and (1)
(1)
Θn+(1) := (x + α0 + β0 ) +
n −1
1
s =0
(n − s)!
(αn(1−)s + βn(1−)s )Dxn−s .
(2.24)
Corollary 2.2. For two different sets of Appell polynomials Rn (x) and Rn (x) and with A1 (t ) and A2 (t ) defined by Eqs. (1.11) and (1.12), let A′1 (t ) A1 (t )
=
∞
αn(1)
n =0
tn n!
A′′1 (t )
,
A1 (t )
=
∞
αn(2)
n =0
tn
(2.25a)
n!
and A′2 (t ) A2 (t )
=
∞
βn(1)
n =0
tn n!
A′′2 (t )
,
A2 (t )
=
∞
βn(2)
n =0
tn n!
,
(2.25b)
respectively. Then, under assumption (2.15), the following recurrence relation for the 2IAP R[n2] (x) holds true: Rn+2 (x) = [2]
(1)
(1)
(2)
(2)
(1) (1)
x2 + 2(α0 + β0 )x + α0 + β0 + 2α0 β0
+
n −1 n s=0
+2
(1)
(1)
(2)
R[n2] (x) (1)
(1)
(1) (1)
2x(αn−s + βn−s ) + αn−s + βn−s + 2(αn−s β0 + α0 βn−s ) R[s2] (x)
s
n n −s−1 n n−s s=0
(2)
s
p=1
p
αn(1−)s−p βp(1) R[s2] (x).
(2.26)
124
S. Khan, M. Riyasat / Journal of Computational and Applied Mathematics 306 (2016) 116–132
The corresponding shift operators are given by 1
Θn(−2) := Φn−−1 Φn− =
n(n − 1)
D2x
(2.27)
and
(1) (1) (2) (2) (1) (1) Θn(+2) := x2 + 2(α0 + β0 )x + α0 + β0 + 2α0 β0 +
n−1
1
s=0
(n − s)!
+2
(1)
(1)
n−s−1
n
1
s=0
(n − s)!
(2)
(2)
(1)
(1)
(1) (1)
2x(αn−s + βn−s )x + αn−s + βn−s + 2(αn−s β0 + α0 βn−s ) Dnx −s
n − s p
p=1
αn(1−)s−p βp(1) Dxn−s .
(2.28)
Remark 2.2. For k = 1 and k = 2, we deduce the following consequences of Theorem 2.2. Corollary 2.3. For n ∈ N, the 2IAP R[n2] (x) satisfy the following differential equation: (x)
Ln,1 {R[n2] (x)} = n R[n2] (x),
(2.29)
where (x)
(1)
(1)
Ln,1 := (x + α0 + β0 )Dx +
n−1
1
s=0
(n − s)!
(αn(1−)s + βn(1−)s )Dxn−s+1 .
(2.30)
Corollary 2.4. For n ∈ N, the 2IAP R[n2] (x) satisfy the following differential equation: (x)
Ln,2 {R[n2] (x)} = (n2 + 3n) R[n2] (x),
(2.31)
where (x)
(1)
(1)
Ln,2 := 4 x + α0 + β0
+
n −1
1
s=0
(n − s)!
+ 2αn(1−)s β0(1)
+
(1)
(1)
(2)
(2)
(1) (1)
Dx + x2 + 2(α0 + β0 )x + α0 + β0 + 2α0 β0 (1)
(1)
4αn−s + 4βn−s Dnx −s+1 +
(1) (1) 2α0 βn−s
Dnx −s+2
+2
n−1
1
s=0
(n − s)!
n
1
s=0
(n − s)!
(1)
(1)
D2x (2)
(2)
2x(αn−s + βn−s ) + αn−s + βn−s
n−s−1
n − s p=1
p
αn(1−)s−p βp(1) Dxn−s+2 .
(2.32)
In the next section, the integral equations for the Appell polynomials Rn (x) and 2IAP R[n2] (x) are derived. 3. Integral equations Integral equations arise in many scientific and engineering problems. A large class of initial and boundary value problems can be converted to Volterra or Fredholm integral equations. The potential theory contributed more than any field to give rise to integral equations. Mathematical physics models, such as diffraction problems scattering in quantum mechanics, conformal mapping and water waves also contributed to the creation of integral equations. The integral equations for the Appell polynomials Rn (x) and 2-iterated Appell polynomials R[n2] (x) have not been found before. Hence, the integral equations satisfied by these polynomials may be used to express the problems arising in new and emerging areas of sciences. We recall that the Appell polynomials Rn (x) are the solutions of the following differential equation [24, p. 234(2.1)]:
αn−1 (n) αn−2 (n−1) α1 y + y + · · · + y′′ + (x + α0 )y′ − ny = 0. (n − 1)! (n − 2)! 1!
(3.1)
First, we derive the integral equation for the Appell polynomials Rn (x) by making use of the differential equation (3.1) by proving the following result:
S. Khan, M. Riyasat / Journal of Computational and Applied Mathematics 306 (2016) 116–132
125
Theorem 3.1. For the Appell polynomials Rn (x) defined by Eq. (1.1), the following homogeneous Volterra integral equation holds true:
x n −3 φ(x) = − (n − 1)δ2 Pn−2 − (n − 1)(n − 2)δ3 Pn−2 x + Pn−3 − · · · − (n − 1)!δn−1 Pn−2 (n − 3)! n −2 n −4 x x x x xn−3 + Pn−3 + · · · + P2 + P1 − (n − 1)! + δn Pn−2 + Pn−3 (n − 4)! 1! αn−1 (n − 2)! (n − 3)! n−1 n−2 n! x x x x Pn−2 + Pn−3 + · · · + mRm−1 + Rm + · · · + P1 + mRm−1 + 1! αn−1 (n − 1)! (n − 2)! 1! x (n − 1)δ2 + (n − 1)(n − 2)δ3 (x − ξ ) + · · · + (n − 1)!δn−1 (x − ξ )n−3 − 0
x n! + (n − 1)! + δn (x − ξ )n−2 − (x − ξ )n−1 φ(ξ )dξ , αn−1 αn−1
(3.2)
α
where δi := α n−i (i = 2, 3, . . . , n − 1, n) and numerical coefficients αk (k = 1, 2, . . . , n − 1) are given by n−1 A′ (t ) A(t )
=
∞
αn
n =0
tn n!
.
(3.3)
Proof. Consider differential equation (3.1) in the following form: dn y dxn
+ (n − 1)δ2
dn−1 y dxn−1
+ (n − 1)(n − 2)δ3
dn−2 y dxn−2
+ · · · + (n − 1)!δn−1
d2 y dx2
+ (n − 1)!
x dy n! × + δn − y = 0, αn−1 dx αn−1
(3.4)
α
where δi := α n−i (i = 2, 3, . . . , n − 1, n) and αk (k = 1, 2, . . . , n − 1) are the numerical coefficients given by Eq. (3.3). n−1 In view of Eqs. (1.1)–(1.3) the initial conditions are obtained as follows: y(0) = Rm , y′ (0) = mRm−1 , y′′ (0) = m(m − 1)Rm−2 = P1 :=
1 (m − k)Rm−2 , k=0
.. . y(n−2) (0) = m(m − 1) · · · (m − n + 3)Rm−n+2 = Pn−3 :=
n −3
(m − k)Rm−n+2 ,
k=0
y(n−1) (0) = m(m − 1) · · · (m − n + 2)Rm−n+1 = Pn−2 :=
n −2
(m − k)Rm−n+1 ,
k=0
where prime denotes differentiation with respect to x and
Pn−s :=
n −s
(m − k)Rm−n+(s−1) ,
s = n − 1, n − 2, . . . , 3, 2.
k=0
Consider dn y dxn
= φ(x).
Integrating the above equation and using initial conditions (3.5), we have dn−1 y dxn−1 dn−2 y dxn−2
x
φ(ξ )dξ + Pn−2 ,
= 0 x
φ(ξ )dξ 2 + Pn−2 x + Pn−3 ,
= 0
(3.5)
126
S. Khan, M. Riyasat / Journal of Computational and Applied Mathematics 306 (2016) 116–132
.. . x x n −3 x n −4 x n−2 φ(ξ ) d ξ + P = + P + · · · + P2 + P1 , n − 2 n − 3 2 dx (n − 3)! (n − 4)! 1! 0 x n−2 n−3 x x x dy φ(ξ )dξ n−1 + Pn−2 = + Pn−3 + · · · + P1 + mRm−1 , dx (n − 2)! (n − 3)! 1! 0 x xn−1 xn−2 x φ(ξ )dξ n + Pn−2 y= + Pn−3 + · · · + mRm−1 + Rm . (n − 1)! (n − 2)! 1! 0 d2 y
(3.6)
Using expressions (3.6) in Eq. (3.4), we find
x x φ(ξ )dξ 2 + Pn−2 x + Pn−3 φ(ξ )dξ + Pn−2 + (n − 1)(n − 2)δ3 φ(x) + (n − 1)δ2 0 0 x n −3 x xn−4 x φ(ξ )dξ n−2 + Pn−2 + · · · + (n − 1)!δn−1 + Pn−3 + · · · + P2 + P1 (n − 3)! (n − 4)! 1! 0 x x n−2 n−3 x x x + (n − 1)! φ(ξ )dξ n−1 + Pn−2 + δn + Pn−3 + · · · + P1 + mRm−1 αn−1 (n − 2)! (n − 3)! 1! 0 n! x n−1 n−2 x x x − φ(ξ )dξ n + Pn−2 + Pn−3 + · · · + mRm−1 + Rm = 0, αn−1 0 (n − 1)! (n − 2)! 1! which after simplification yields assertion (3.2).
Theorem 3.2. For the 2IAP R[n2] (x) defined by Eq. (1.10), the following homogeneous Volterra integral equation holds true:
ψ(x) = −(n − 1)γ2
n−2
(m − k)T(n−1)−p
− (n − 1)(n − 2)γ3
k=0
+
n−3
(m − k)T(n−1)−p x
k=0
(m − k)T(n−2)−p − · · · − (n − 1)!γn−1
k=0
n−2
n−2 k=0
(m − k)T(n−1)−p
xn−3 + ··· (n − 3)!
2 1 n−2 x x + (m − k)T3−p + (m − k)T2−p − (n − 1)! + γn (m − k) 1! αn−1 + βn−1 k=0 k=0 k=0 1 n −2 x xn−2 n! (m − k)T2−p + ··· + + mT1−p + (m − k) (n − 2)! 1! αn−1 + βn−1 k=0 k=0 xn−1 x x × T(n−1)−p + · · · + mT1−p + T −p − (n − 1)γ2 + (n − 1)(n − 2) (n − 1)! 1! 0 x × γ3 (x − ξ ) + · · · + (n − 1)!γn−1 (x − ξ )n−3 + (n − 1)! + γn (x − ξ )n−2 αn−1 + βn−1 n! − (x − ξ )n−1 ψ(ξ )dξ , αn−1 + βn−1
× T(n−1)−p
α
(3.7)
+β
where γi := α n−i +βn−i (i = 2, 3, . . . , n − 1, n) and the numerical coefficients αk , βk (k = 1, 2, . . . , n − 1) are given by n−1 n−1 A′1 (t ) A1 (t )
=
∞
αn
n =0
tn
(3.8a)
n!
and A′2 (t ) A2 (t ) respectively.
=
∞ n =0
βn
tn n!
,
(3.8b)
S. Khan, M. Riyasat / Journal of Computational and Applied Mathematics 306 (2016) 116–132
127
Proof. Consider differential equation (2.29) (with (2.30)) for the 2IAP R[n2] (x) in the following form: dn−1 y
dn y
d n −2 y
d2 y
+ (n − 1)γ2 n−1 + (n − 1)(n − 2)γ3 n−2 + · · · + (n − 1)!γn−1 2 + (n − 1)! dx dx dx dy x n! × + γn − y = 0, αn−1 + βn−1 dx αn−1 + βn−1
dxn
α
(3.9)
+β
where γi := α n−i +βn−i (i = 2, 3, . . . , n − 1, n) and αk , βk (k = 1, 2, . . . , n − 1) are the numerical coefficients given by Eqs. n−1 n−1 (3.8a) and (3.8b), respectively. In view of Eqs. (1.10)–(1.14), the initial conditions are obtained as follows:
m m
y(0) =
p
p=0
Rm−p Rp = T−p ,
m−1
y′ (0) = m
m − 1 p
p=0
Rm−1−p Rp = mT1−p ,
m−2
y′′ (0) = m(m − 1)
m − 2 p
p=0
Rm−2−p Rp =
1 (m − k)T2−p ,
(3.10)
k=0
.. . y(n−2) (0) = m(m − 1) · · · (m − n + 3)
m−n+2
m − n + 2 p
p=0
y(n−1) (0) = m(m − 1) · · · (m − n + 2)
Rm−n+2−p Rp =
m − n + 1 p
(m − k)T(n−2)−p ,
k=0
m−n+1
p=0
n−3
Rm−n+1−p Rp =
n−2
(m − k)T(n−1)−p ,
k=0
where prime denotes differentiation with respect to x and
Ts−p :=
m−s m−s p
p=0
Rm−s−p Rp ,
s = 0, 1, 2, . . . , n − 2, n − 1.
Consider dn y
= ψ(x), x n −2 ( m − k ) T = ψ(ξ ) d ξ + , ( n − 1 )− p n −1
dxn dn−1 y dx
0
n −2
d
y
=
dxn−2
k=0 x
n −2 n −3 ψ(ξ )dξ 2 + (m − k)T(n−1)−p x + (m − k)T(n−2)−p
0
k=0
.. .
n −2
k =0
2 1 xn−3 x + · · · + ( m − k ) T + ( m − k ) T , 3 − p 2 − p dx2 (n − 3)! 1! 0 k=0 k=0 k=0 x n−2 1 x n −2 x dy = ψ(ξ )dξ n−1 + (m − k)T(n−1)−p + ··· + (m − k)T2−p + mT1−p , dx (n − 2)! 1! 0 k=0 k=0 x n−2 x xn−1 y= ψ(ξ )dξ n + (m − k)T(n−1)−p + · · · + mT1−p + T −p . (3.11) (n − 1)! 1! 0 k=0
d2 y
=
x
ψ(ξ )dξ n−2 +
(m − k)T(n−1)−p
Using expressions (3.11) in Eq. (3.9), we find n −2 x x ψ(x) + (n − 1)γ2 ψ(ξ )dξ + (m − k)T(n−1)−p + (n − 1)(n − 2)γ3 ψ(ξ )dξ 2 0
+
n −2 k =0
0
k=0
(m − k)T(n−1)−p x +
n −3 k=0
(m − k)T(n−2)−p
x + · · · + (n − 1)!γn−1 ψ(ξ )dξ n−2 0
128
S. Khan, M. Riyasat / Journal of Computational and Applied Mathematics 306 (2016) 116–132 n −2
2 1 x n −3 x + ··· + (m − k)T3−p + (m − k)T2−p (n − 3)! 1! k=0 k=0 k=0 n − 2 xn−2 x x + (n − 1)! ψ(ξ )dξ n−1 + (m − k)T(n−1)−p + γn + ··· αn−1 + βn−1 (n − 2)! 0 k =0 n−2 1 x x n! − ψ(ξ )dξ n + (m − k)T(n−1)−p + (m − k)T2−p + mT1−p 1! αn−1 + βn−1 0 k=0 k=0 n −1 x x × = 0, + · · · + mT1−p + T −p (n − 1)! 1!
+
(m − k)T(n−1)−p
which after simplification yields assertion (3.7).
(3.12)
Remark 3.1. We have given an independent proof to derive integral equation (3.2) for the Appell polynomials Rn (x). We note that integral equation (3.2) can also be obtained from integral equation (3.7) of the 2IAP R[n2] (x) by taking p = 0 and α βn−i = 0 (i = 1, 2, 3, . . . , n − 1, n) i.e. replacing γi by α n−i := δi (i = 2, 3, . . . , n − 1, n). n−1
4. Example (λ)
The recurrence relation, shift operators and differential equation for the generalized Bernoulli polynomials Bn (x) and Bernoulli polynomials Bn (x) are derived in [32]. In this section, the recurrence relation, shift operators, differential equation (λ)[2] (x) are derived. and integral equation for the 2IGBP Bn For this, we consider the following example: Example 4.1. Taking A1 (t ) = A2 (t ) =
λ
t et −1
(λ)[2] (that is when the 2IAP Rn[2] (x) reduce to the 2IGBP Bn (x)) and using
generating function (1.4), we find A′1 (t ) A1 (t )
=
A′2 (t ) A2 (t )
= −λ
∞ Bn+1 (1) t n n=0
n + 1 n!
.
(4.1)
Again, in view of Eqs. (2.21a) and (2.21b), we have ∞
αn
n =0
tn n!
=
∞
βn
n =0
tn n!
= −λ
∞ Bn+1 (1) t n n=0
n + 1 n!
,
which on taking n = 0 and using the value B1 (1) =
(4.2) 1 2
gives
λ α0 = β0 = − .
(4.3)
2
Now, substituting the values of the coefficients from Eqs. (4.2) and (4.3) in Eq. (2.22), we find the following recurrence (λ)[2] relation for the 2IGBP Bn (x): (λ)[2]
2] (x) − 2λ Bn+1 (x) = (x − λ)B(λ)[ n
n−1 n Bn−s+1 (1) s=0
s
n−s+1
2] B(λ)[ (x). s
(4.4) (λ)[2]
Similarly, from Eqs. (2.23) and (2.24), we find the following shift operators for the 2IGBP Bn
Θn−(1) := Φn− =
1 n
(x):
Dx
(4.5)
and
Θn+(1) := (x − λ) − 2λ
n−1 Bn−s+1 (1) n−s D . (n − s + 1)! x s=0
(4.6) (λ)[2]
Further, from Eqs. (2.31) and (2.32), the following differential equation for the 2IGBP Bn n −1 Bn−s+1 (1) n−s+1 (x − λ)Dx − 2λ Dx − n Bn(λ)[2] (x) = 0. (n − s + 1)! s =0
(x) is obtained: (4.7)
S. Khan, M. Riyasat / Journal of Computational and Applied Mathematics 306 (2016) 116–132
129
Remark 4.1. Taking λ = 1 in Eqs. (4.5)–(4.7) and in view of relation (1.9), the recurrence relation, shift operators and differential equation for the 2IBP B[n2] (x) are obtained as: Bn+1 (x) = (x − 1)B[n2] (x) − 2 [2]
n −1 n Bn−s+1 (1) s=0
Θn−(1) := Φn− =
1 n
s
n−s+1
B[s2] (x),
(4.8)
Dx ;
(4.9)
n −1 Bn−s+1 (1) n−s D ( n − s + 1)! x s=0
Θn+(1) := (x − 1) − 2
(4.10)
and
(x − 1)Dx − 2
n−1 Bn−s+1 (1) n−s+1 Dx − n B[n2] (x) = 0, (n − s + 1)! s=0
(4.11)
respectively. (λ)
Next, we derive the integral equation for the generalized Bernoulli polynomials Bn (x) from the integral equation of the Appell polynomials Rn (x).
Taking A(t ) =
t et −1
λ
, (that is when the Appell polynomials Rn (x) reduce to the generalized Bernoulli polynomials
(λ)
Bn (x)) and using generating function (1.4), we find A′ (t ) A(t )
= −λ
∞ Bn+1 (1) t n
n + 1 n!
n =0
.
(4.12)
Again, in view of Eq. (2.21a), we have ∞ n =0
αn
tn n!
= −λ
∞ Bn+1 (1) t n
n + 1 n!
n =0
.
(4.13)
Taking n = 0, 1, . . . , n − 2, n − 1 in Eq. (4.13) and equating the coefficients of the same powers of t, we find
α0 = −λB1 (1) = − α1 = −λ
B2 (1) 2
.. . αn−3 = −λ αn−2 = −λ αn−1 = −λ
1
2
2
=−
Bn−2 (1) n−2 Bn−1 (1) n−1 Bn (1) n
λ
since B1 (1) =
λ
12
since B2 (1) =
, 1 6
,
, ,
.
(4.14) α
Using the values from Eq. (4.14), we obtain the coefficients δi := α n−i (i = 2, 3, . . . , n − 1, n). Finally, substituting these n−1 values in integral equation (3.2) of the Appell polynomials Rn (x) and replacing the coefficients Rm of the Appell numbers (λ) by the coefficients Bm of the generalized Bernoulli numbers, the following integral equation for the generalized Bernoulli (λ) polynomials Bn (x) is obtained:
B (1) (1) n! 1 n −2 Pn−2 − n(n − 1) Pn−2 x + Pn−3 − · · · − Bn (1) Bn (1) 12 Bn (1) xn−3 xn−4 x x 1 1 × Pn−2 + Pn−3 + · · · + P2 + P1 + n! − (n − 3)! (n − 4)! 1! λ 2 Bn (1) xn−2 xn−3 x n! n × Pn−2 + Pn−3 + · · · + P1 + mB(λ) m−1 − (n − 2)! (n − 3)! 1! λ Bn (1)
φ(x) = −n
B
n −1
130
S. Khan, M. Riyasat / Journal of Computational and Applied Mathematics 306 (2016) 116–132
xn−1 x n −2 x B (1) x n−1 (λ) n + Pn−3 + · · · + mB(λ) + B − m m−1 (n − 1)! (n − 2)! 1! Bn (1) 0 x B (1) (n − 1)! 1 1 n−2 (x − ξ ) + · · · + (x − ξ )n−3 − (n − 1)! − + (n − 1) Bn (1) 12 Bn (1) λ 2 1 n! 1 × (x − ξ )n−2 + (x − ξ )n−1 φ(ξ )dξ , Bn (1) λ Bn (1) ×
Pn−2
(4.15)
where
Pn−s :=
n −s
(m − k)B(λ) m−n+(s−1) ,
s = n − 1, n − 2, . . . , 3, 2.
k=0
(λ)[2]
Further, we derive the integral equation for the 2-iterated generalized Bernoulli polynomials Bn equation of the 2IAP R[n2] (x).
Taking A1 (t ) = A2 (t ) =
t et −1
λ
(x) from the integral
(λ)[2] , (that is when the 2IAP R[n2] (x) reduce to the 2IGBP Bn (x)) and using generating
function (1.4), we find A′1 (t ) A1 (t )
A′2 (t )
=
A2 (t )
∞ Bn+1 (1) t n
= −λ
n + 1 n!
n=0
.
(4.16)
Again, in view of Eqs. (2.21a) and (2.21b), we have ∞ n =0
αn
tn n!
=
∞
βn
n =0
tn
= −λ
n!
∞ Bn+1 (1) t n n=0
n + 1 n!
.
(4.17)
Taking n = 0, 1, . . . , n − 2, n − 1 in Eq. (4.17) and equating the coefficients of the same powers of t, we find
α0 = β0 = −λB1 (1) = − α1 = β1 = −λ
B2 (1) 2
.. . αn−3 = βn−3 = −λ αn−2 = βn−2 = −λ αn−1 = βn−1 = −λ
λ 2
=−
λ
12
Bn−2 (1) n−2 Bn−1 (1) n−1 Bn (1) n
since B1 (1) =
1
, 1 since B2 (1) = , 2
6
, ,
.
(4.18) α
+β
Making use of the values from Eq. (4.18), we obtain the coefficients γi := α n−i +βn−i (i = 2, 3, . . . , n − 1, n). Finally, n−1 n−1 substituting these values in integral equation (3.7) of the 2IAP R[n2] (x) and replacing the Appell coefficients Rm , Rp by the (λ)
(λ)
coefficients Bm , Bp
of the generalized Bernoulli numbers, the following integral equation for the 2-iterated generalized (λ)[2]
(x) is obtained: n −2 n −2 B (1) B (1) n −2 n −1 ψ(x) = −n (m − k)T(n−1)−p − n(n − 1) (m − k)T(n−1)−p x Bn (1) Bn (1) k=0 k=0
Bernoulli polynomials Bn
+
n −3 n −2 x n −3 n! 1 (m − k)T(n−2)−p − · · · − (m − k)T(n−1)−p + ··· 12 Bn (1) (n − 3)! k=0 k=0
+
2 1 n−2 x x 1 1 (m − k)T3−p + (m − k)T2−p + n! − (m − k)T(n−1)−p 1! 2λ 2 Bn (1) k=0 k=0 k=0
1 n −2 x n! n + mT1−p − (m − k)T2−p (m − k)T(n−1)−p (n − 2)! 1! 2λ Bn (1) k=0 k=0 x B (1) x xn−1 Bn−1 (1) n −2 × + · · · + mT1−p + T−p − n + (n − 1) (x − ξ ) + · · · (n − 1)! 1! Bn (1) Bn (1) 0
×
xn−2
+ ··· +
S. Khan, M. Riyasat / Journal of Computational and Applied Mathematics 306 (2016) 116–132
131
x n! 1 (n − 1)! 1 1 1 + (x − ξ )n−3 − (n − 1)! (x − ξ )n−2 + − 12 Bn (1) 2λ 2 Bn (1) 2λ Bn (1) n −1 ψ(ξ )dξ , × (x − ξ )
(4.19)
where
Ts−p :=
m−s m−s p=0
p
(λ)
Bm−s−p B(λ) p ,
s = 0, 1, 2, . . . , n − 2, n − 1.
(λ)
Remark 4.2. By taking λ = 1 in integral equations (4.15) and (4.19) of the generalized Bernoulli polynomials Bn (x) and (λ)[2] 2IGBP Bn (x), the integral equations of the Bernoulli polynomials Bn (x) and 2IBP B[n2] (x) can be derived. 5. Concluding remarks The orthogonal polynomials in general and the classical orthogonal polynomials in particular have been the object of extensive works. They are connected with numerous problems of applied mathematics, theoretical physics, chemistry, approximation theory and several other mathematical branches. In particular, their applications are being widely used in theories as Padé approximants, continued fractions, spectral study of Schrödinger discrete operators, polynomial solutions of second-order differential equations and others. The notions of d-dimensional orthogonality for polynomials [33], vectorial orthogonality [34] or simultaneous orthogonality [35] are the generalizations of ordinary orthogonality for polynomials. Such polynomials are characterized by the fact that they satisfy a d + 1-order recurrence relationship, that is a relation between d + 2 consecutive polynomials [34]. All these new notions of d-orthogonality for polynomials and, equivalently, 1/d-orthogonality [36] appear as particular cases of the general notion of biorthogonality studied in [37]. Recently they have been the subject of numerous investigations and applications. In particular, they are connected with the study of vector Padé approximants [34,38], simultaneous Padé approximants [35] and other problems such as vectorial continued fractions, polynomials solutions of the higher-order differential equations, spectral study of multi diagonal non-symmetric operators [39]. In this article, a set of finite order recurrence relations and differentials equations for the Appell and 2-iterated Appell polynomials Rn (x) and R[n2] (x), respectively are established. The integral equations for the Appell and 2-iterated Appell polynomials Rn (x) and R[n2] (x), respectively are also derived. Since, the Appell and 2-iterated Appell polynomials are important from the point of view of their applications in various fields, therefore the differential and integral equations satisfied by these polynomials may be used to solve the existing as well as new emerging problems in certain branches of science. The problems of finding all polynomial sequences, which are at the same time Appell polynomials and d-orthogonal is considered in [23]. The recurrence relation and differential equations of the 2IAP A[n2] (x) established in this article may be used to study the d-orthogonality of these polynomials. This aspect may be considered in further investigation. Acknowledgments The authors are thankful to the reviewer for several useful comments and suggestions towards the improvement of this paper. This work has been done under Senior Research Fellowship (Award letter No. F1-17.1/2012–13, MANF-MUS-UTT-9243) awarded to the second author by the University Grants Commission, Government of India, New Delhi. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
P. Appell, Sur une classe de polynômes, Ann. Sci. École Norm. Sup. 9 (2) (1880) 119–144. S. Roman, The Umbral Calculus, Academic Press, New York, 1984. H.T. Davis, The Theory of Linear Operator, Principia Press, Bloomington, Indiana, 1936. A. Di Bucchianico, D. Loeb, A selected survey of umbral calculus, Electron. J. Combin. DS3 (2000) 1–34. A. Erdélyi, W. Magnus, F. Oberhettinger, F.G. Tricomi, Higher Transcendental Functions, Vol. I, McGraw-Hill Book Company, New York, Toronto, London, 1953. A. Erdélyi, W. Magnus, F. Oberhettinger, F.G. Tricomi, Higher Transcendental Functions, Vol. III, McGraw-Hill Book Company, New York, Toronto, London, 1955. E.D. Rainville, Special Functions, Reprint of 1960 First Edition, Chelsea Publishing Company, Bronx, New York, 1971. J. Stoer, Introduzione all’Analisi Numerica, Zanichelli, Bologna, 1972. Subuhi Khan, N. Raza, 2-Iterated Appell polynomials and related numbers, Appl. Math. Comput. 219 (17) (2013) 9469–9483. Subuhi Khan, M. Riyasat, A study on the 2-Iterated q-Appell polynomials from determinantal point of view, J. Math. Anal. Appl. (2016) submitted for publication. T.H. Cormen, C.E. Leiserson, R.L. Rivest, C. Stein, Introduction to Algorithms, MIT Press, 2009. R. Sedgewick, F. Flajolet, An Introduction to the Analysis of Algorithms, Addison-Wesley, 2013. J. Thomas Sargent, Dynamic Macroeconomic Theory, Harvard University Press, 1987.
132 [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39]
S. Khan, M. Riyasat / Journal of Computational and Applied Mathematics 306 (2016) 116–132 W. Johnson, A Treatise on Ordinary and Partial Differential Equations, John Wiley and Sons, 1913, in University of Michigan Historical Math Collection. R.P. Boas, R.C. Buck, Polynomial Expansions of Analytic Functions, Springer-Verlag, Berlin, Gottingen, Heidelberg, 1958. I.M. Sheffer, A differential equation for Appell polynomials, Bull. Amer. Math. Soc. 41 (12) (1935) 914–923. J. Shohat, The relation of the classical orthogonal polynomials to the polynomials of Appell, Amer. J. Math. 58 (3) (1936) 453–464. G. Bretti, C. Cesarano, P.E. Recci, Laguerre-type exponential and generalized Appell polynomials, Comput. Math. Appl. 48 (5–6) (2004) 833–839. G. Bretti, M.X. He, P.E. Ricci, On quadrature rules associated with Appell polynomials, Int. J. Appl. Math. 11 (2002) 1–14. G. Bretti, P. Natalini, P.E. Ricci, Generalizations of the Bernoulli and Appell polynomials, Abstr. Appl. Anal. 7 (2004) 613–623. G. Bretti, P.E. Ricci, Multidimensional extensions of the Bernoulli and Appell polynomials, Taiwanese J. Math. 8 (2004) 415–428. G. Dattoli, P.E. Ricci, C. Cesarano, Differential equations for Appell type polynomials, Fract. Calc. Appl. Anal. 5 (2002) 69–75. K. Douak, The relation of the d-orthogonal polynomials to the Appell polynomials, J. Comput. Appl. Math. 70 (2) (1996) 279–295. M.X. He, P.E. Ricci, Differential equation of Appell polynomials via the factorization method, J. Comput. Appl. Math. 139 (2002) 231–237. M.E.H. Ismail, Remarks on differential equation of Appell polynomials via the factorization method, J. Comput. Appl. Math. 154 (2003) 243–245. M.A. Özarslan, B. Yilmaz, A set of finite order differential equations for the Appell polynomials, J. Comput. Appl. Math. 259 (2014) 108–116. 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. J.F. Steffensen, The poweriod, an extension of the mathematical notion of power, Acta Math. 73 (1941) 333–366. G. Dattoli, C. Cesarano, D. Sacchetti, A note on the monomiality principle and generalized polynomials, J. Math. Anal. Appl. 227 (1997) 98–111. L. Infeld, T.E. Hull, The factorization method, Rev. Modern Phys. 23 (1951) 21–68. A.J. Jerry, Introduction to Integral Equations with Applications (Second Edition), John Wiley and Sons, Inc., New York, Chichester, Weinheins, Brisbane, Singapore, Toronto, 1999. Da-Qian Lu, Some properties of Bernoulli polynomials and their generalizations, Appl. Math. Lett. 24 (2011) 746–751. P. Maroni, L’orthogonalité et les récurrences de polynoˆ mes d’ordre supérieur á deux, Ann. Fac. Sci. Toulouse 10 (1989) 105–139. J. Van Iseghem, Approximants de padé vectoriels (Thése d’état), Univ. des Sciences et Techniques de Lille-Flandre-Artois, 1987. M.G. de Bruin, Simultaneous padé approximation and orthogonality, in: Lectures Notes in Math., Vol. 1171, Springer, Berlin, 1985, pp. 74–83. A. Boukhemis, P. Maroni, Une caractérisation des polynoˆ mes strictement 1/p-orthogonaux de type Sheffer, étude du cas p = 2, J. Approx. Theory 54 (1988) 67–91. C. Brezinski, Biorthogonality and its Applications to Numerical Analysis, Marcel Dekker, New York, 1992. J. Van Iseghem, Vector orthogonal relations, vector QD-algorithm, J. Comput. Appl. Math. 19 (1987) 141–150. A.I. Aptekarev, V.A. Kaliaguine, W. Van Assche, Criterion for the resolvent set of non-symmetric tridiagonal operators, Proc. Amer. Math. Soc. 123 (8) (1995) 2423–2430.