Monomiality principle, operational methods and family of Laguerre–Sheffer polynomials

J. Math. Anal. Appl. 387 (2012) 90–102

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Monomiality principle, operational methods and family of Laguerre–Sheffer polynomials ✩ Subuhi Khan ∗ , Nusrat Raza Department of Mathematics, Aligarh Muslim University, Aligarh, India

a r t i c l e

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a b s t r a c t

Article history: Received 30 March 2011 Available online 30 August 2011 Submitted by B.C. Berndt

In this paper, the Laguerre–Sheffer polynomials are introduced by using the monomiality principle formalism and operational methods. The generating function for the Laguerre– Sheffer polynomials is derived and a correspondence between these polynomials and the Sheffer polynomials is established. Further, differential equation, recurrence relations and other properties for the Laguerre–Sheffer polynomials are established. Some concluding remarks are also given. © 2011 Elsevier Inc. All rights reserved.

Keywords: Sheffer polynomials Laguerre–Sheffer polynomials Laguerre–Appell polynomials Monomiality principle Operational methods

1. Introduction and preliminaries Sequences of polynomials play an important role in various branches of science. One of the important classes of polynomial sequences is the class of Sheffer sequences. There are several ways to define this class, 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 [18, p. 222 (Theorem 72)] (which we shall hereafter call Sheffer-type), if sn (x) possesses the exponential generating function of the form





A (t ) exp xH (t ) =

∞ 

sn (x)

n =0

tn n!

,

(1.1)

where A (t ) and H (t ) have (at least the formal) expansions:

A (t ) =

∞  n =0

An

tn n!

,

A 0 = 0

(1.2a)

,

H 1 = 0,

(1.2b)

and

H (t ) =

∞  n =1

Hn

tn n!

respectively. ✩ This work has been done under a Major Research Project No. F.33-110/2007 (SR) sanctioned to the first author by the University Grants Commission, Government of India, New Delhi. Corresponding author. E-mail address: [email protected] (S. Khan).

*

0022-247X/$ – see front matter doi:10.1016/j.jmaa.2011.08.064

©

2011 Elsevier Inc. All rights reserved.

S. Khan, N. Raza / J. Math. Anal. Appl. 387 (2012) 90–102

91

The Sheffer class contains important sequences such as the Hermite, Laguerre, Bessel, Bernoulli, Poisson–Charlier polynomials etc. (A table containing some known Sheffer polynomials is given in Appendix A.) These polynomials are important from the viewpoint of applications in physics and number theory. Properties of Sheffer sequences are naturally handled within the framework of modern classical umbral calculus by Roman [19]. We recall the following result [19, p. 17], which can be viewed as an alternate definition of Sheffer sequences: Let f (t ) be a delta series and let g (t ) be an invertible series of the following form:

f (t ) =

∞ 

fn

n =0

tn n!

,

f 0 = 0, f 1 = 0

(1.3a)

,

g 0 = 0.

(1.3b)

and

g (t ) =

∞ 

gn

n =0

tn n!

Then there exists a unique sequence sn (x) of polynomials satisfying the orthogonality conditions







g (t ) f (t )k  sn (x) = n!δn,k ,

for all n, k  0.

(1.4)

According to Roman [19, p. 18 (Theorem 2.3.4)] the polynomial sequence sn (x) is uniquely determined by two (formal) power series given by Eqs. (1.3a) and (1.3b). The exponential generating function of sn (x) is then given by



1 g ( f −1 (t ))



exp xf −1 (t ) =

∞ 

sn (x)

n =0

tn n!

,

(1.5)

for all x in C, where f −1 (t ) is the compositional inverse of f (t ). In view of Eqs. (1.1) and (1.5), we have

A (t ) =

1

(1.6a)

g ( f −1 (t ))

and

H (t ) = f −1 (t ).

(1.6b)

The sequence sn (x) in Eq. (1.4) is the Sheffer sequence for the pair ( g (t ), f (t )). The Sheffer sequence for (1, f (t )) is called the associated sequence for f (t ) and the Sheffer sequence for ( g (t ), t ) is called the Appell sequence for g (t ) [19, p. 17]. Roman [19] characterized Appell sequences in several ways. We recall that the Appell sets may be defined by either of the following equivalent conditions [20, p. 398]: { A n (x)} (n ∈ N), is an Appell set ( A n being of degree exactly n) if either d (i) dx an (x) = nan−1 (x) (n ∈ N), or (ii) there exists an exponential generating function of the form

A (t ) exp(xt ) =

∞ 

an (x)

n =0

tn n!

,

(1.7)

where A (t ) has (at least the formal) expansion (1.2a). From Eq. (1.6a), we have for the Appell sequence

A (t ) =

1 g (t )

(1.8)

.

In view of Eq. (1.8), the generating function (1.7) may also be expressed as:

1 g (t )

exp(xt ) =

∞ 

an (x)

n =0

tn n!

.

(1.9)

Also, we note that for H (t ) = t, generating function (1.1) of the Sheffer polynomials sn (x) reduces to generating function (1.7) of the Appell polynomials an (x). Moreover, according to Sheffer identity [19, p. 21 (Theorem 2.3.9)], a sequence sn (x) is Sheffer for ( g (t ), f (t )) for some invertible g (t ), if and only if

sn (x + y ) =

n   n k =0

k

pk ( y )sn−k (x),

(1.10)

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S. Khan, N. Raza / J. Math. Anal. Appl. 387 (2012) 90–102

for all y in C, where pn (x) is associated to f (t ). Also, if sn (x) is Sheffer for ( g (t ), f (t )), then the recurrence relation for sn (x) is given by [19, p. 50]



sn+1 (x) = x −

g  (t )



1 f  (t )

g (t )

sn (x).

(1.11)

Recently, it has been shown that the Sheffer polynomials and the monomiality principle, along with the underlying operational formalism provide a powerful tool for the investigation of the properties of a wide class of polynomials, see for example [8,17]. The idea of monomiality traces back to the early 1940s, when J.F. Steffensen [21] suggested the concept of poweroid. The concept of monomiality principle is reformulated and developed by Dattoli [4], according to which, the ˆ and Pˆ playing, respectively, the role of polynomial set pn (x)n∈N is “quasi-monomial” provided there exist two operators M multiplicative and derivative operators, for the family of polynomials. These operators satisfy the following identities, for all n ∈ N:









ˆ pn (x) = pn+1 (x) M and

(1.12)

Pˆ pn (x) = npn−1 (x).

(1.13)

ˆ and Pˆ also satisfy the commutation relation: The operators M

ˆ −M ˆ ] = Pˆ M ˆ Pˆ = 1ˆ , [ Pˆ , M

(1.14)

and thus display the Weyl group structure. If the considered polynomial set pn (x)n∈N is quasi-monomial, its properties can ˆ and Pˆ operators. In fact: easily be derived from those of the M (i) Combining the recurrences (1.12) and (1.13), we have





ˆ Pˆ pn (x) = npn (x), M

(1.15)

ˆ and Pˆ have a differential realization. which can be interpreted as the differential equation satisfied by pn (x), if M (ii) Assuming here and in the sequel p 0 (x) = 1, then pn (x) can be explicitly constructed as:

ˆ n {1}, pn (x) = M

(1.16)

which yields the series definition for pn (x). (iii) Last identity (1.16) implies that the exponential generating function of pn (x) can be given in the form:

ˆ ){1} = exp (t M

∞  ˆ )n (t M

n!

n =0

{1} =

∞ n  t n =0

n!

ˆ n {1} = M

∞ n  t n =0

n!

pn (x)

and therefore

ˆ ){1} = exp (t M

∞ n  t n =0

n!

pn (x).

(1.17)

ˆ and Pˆ . It has been shown in [17], that if sn (x) are of Sheffer-type then it is possible to give explicit representations of M ˆ ( X , D ) and Pˆ = Pˆ ( D ), then sn (x) satisfying Eqs. (1.12) and (1.13) are necessarily of Sheffer-type. The ˆ =M Conversely if M raising and lowering operators for the Sheffer polynomials sn (x) are given by

ˆs= M



X−

g(D )



1 f (D )

g(D)

(1.18a)

,

or, equivalently





ˆ s = X H  H −1 ( D ) + M

A  ( H −1 ( D )) A ( H −1 ( D ))

,

(1.18b)

and

Pˆ s = f ( D ),

(1.19a)

or, equivalently

Pˆ s = H −1 ( D ),

(1.19b)

S. Khan, N. Raza / J. Math. Anal. Appl. 387 (2012) 90–102

93

ˆ s only linearly and the order of X and D in M ˆ s ( X , D ) matters. By a direct calculation respectively. We note that X enters M ˆ s , Pˆ s from Eqs. (1.18a) or (1.18b) and (1.19a) or (1.19b) satisfies commutation relation (1.14), one may check that any pair M see for details [17,19]. The raising and lowering operators for the Appell polynomials an (x) are given by

ˆa=X− M

g(D ) g(D)

,

(1.20a)

,

(1.20b)

or, equivalently

A(D )

ˆa=X+ M

A(D )

and

Pˆ a = D ,

(1.21)

respectively. Dattoli and Torre [11,12] introduced and discussed a theory of two variable Laguerre polynomials (2VLP). The reason of interest for this family of Laguerre polynomials is because of their intrinsic mathematical importance and the fact that these polynomials are shown to be natural solutions of a particular set of partial differential equations which often appear in the treatment of radiation physics problems such as the electromagnetic wave propagation and quantum beam life-time in storage rings [23]. The 2VLP L n (x, y ) are indeed quasi-monomial under the action of the operators [4, p. 149]: 1 ˆ L = y − D− M x

(1.22)

and

Pˆ L = −

∂ ∂ x , ∂x ∂x

(1.23)

respectively, where





n D− f (x) = x

x

1

(n − 1)!

(x − ξ )n−1 f (ξ ) dξ ;

n D− x {1} =

0

xn n!

.

(1.24)

The generating function for L n (x, y ) is given as [4, p. 150]:

exp( yt )C 0 (xt ) =

∞ 

L n (x, y )

n =0

tn n!

(1.25)

,

where C 0 (x) denotes the 0th-order Bessel–Tricomi function and given by the following operational definition:





1 {1}. C 0 (α x) = exp −α D − x

(1.26)

The Bessel–Tricomi function of order n is specified by means of the generating function

 exp t −

x

=

t

∞ 

C n (x)t n ,

(1.27)

n =0

for t = 0 and for all finite x. The Bessel–Tricomi function C n (x) is also defined by the following series [4, p. 150]:

√

n

C n (x) = x− 2 J n (2 x ) =

∞  (−1)k xk , k! (n + k)!

n = 0, 1 , 2 , . . . ,

(1.28)

k =0

with J n (x) being the ordinary cylindrical Bessel function of first kind [1]. The series definition for L n (x, y ) is given as [4, p. 148]:

L n (x, y ) = n!

n  (−1)k yn−k xk k =0

(n − k)!(k!)2

.

(1.29)

Also, the differential equation satisfied by L n (x, y ) is [4, p. 455 (14)]:



xy

∂ ∂2 − ( x − y ) + n L n (x, y ) = 0. ∂x ∂ x2

(1.30)

Further, since

∂ L n (x, y ) = nL n−1 (x, y ), ∂y

(1.31)

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S. Khan, N. Raza / J. Math. Anal. Appl. 387 (2012) 90–102

it follows that L n (x, y ) are the natural solutions of the following partial differential equation:

∂ ∂ ∂ L n (x, y ) = − x L n (x, y ), ∂y ∂x ∂x (−1)n xn , L n (x, 0) = n!

(1.32)

which is a kind of heat diffusion equation. The differential equation (1.32) gives the following operational definitions for L n (x, y ):



1 L n (x, y ) = exp − D − x

and

 L n (x, y ) = exp − y

or, equivalently





∂ −1

∂ Dx



L n (x, y ) = exp − y

∂ n y ∂y



∂ 1 ∂ D− x

(1.33)

1 −D− x

n  ,

(1.34a)

 (−x)n , n!

(1.34b)

where [10, p. 32 (8)]

∂ −1

∂ Dx

∂ ∂ x . ∂x ∂x

=

(1.35)

Operational methods can be exploited to simplify the derivation of the properties associated with ordinary and generalized special polynomials. On the other hand, they may provide a fairly unique method to treat various polynomials from a unified point of view. In this paper, the Laguerre–Sheffer polynomials are introduced and a correspondence between the families of Sheffer and the Laguerre–Sheffer polynomials is established, which is used to derive new identities and results for the Laguerre–Sheffer polynomials. Further, some results for multiplicative and derivative operators and differential equations for the Laguerre–Sheffer and the Laguerre–Appell polynomials are proved. Some concluding remarks are also given. 2. The Laguerre–Sheffer family In view of operational definition (1.33) of 2VLP L n (x, y ) and Eq. (1.1), let the generating function for the Laguerre–Sheffer polynomials L sn (x, y ) be



−1

exp − D x

∞    ∂

tn , A (t ) exp y H (t ) = L sn (x, y ) ∂y n!

(2.1)

n =0

which on using the shift identity [9, p. 4]



 ∂ ∂ , exp λ f (x) = f (x + λ) exp λ ∂x ∂x

(2.2)

takes the form







1 A (t ) exp y − D − H (t ) {1} = x

∞ 

L sn (x,

y)

n =0

tn n!

(2.3)

.

Again, in view of operational definition (1.34a), we can write the following equivalent form of Eq. (2.1)

 exp − y

∂



−1

∂ Dx







1 A (t ) exp − D − x H (t )

=

∞ 

L sn (x,

y)

n =0

tn n!

,

(2.4)

which on using identity (2.2), gives Eq. (2.3). Now, decoupling the exponential operator in the r.h.s. of Eq. (2.3), by using the Weyl identity [9, p. 7] ˆ

ˆ

ˆ

ˆ

[ Aˆ , Bˆ ] = k,

e A + B = e −k/2 e A e B ,

(2.5)

we find









1 {1} = A (t ) exp y H (t ) exp − H (t ) D − x

∞  n =0

L sn (x,

y)

tn n!

,

(2.6)

S. Khan, N. Raza / J. Math. Anal. Appl. 387 (2012) 90–102

95

which on using Eq. (1.26) finally yields the generating function for the Laguerre–Sheffer polynomials L sn (x, y ) in the following form:









A (t ) exp y H (t ) C 0 xH (t ) =

∞ 

L sn (x,

y)

n =0

tn n!

(2.7a)

,

which in view of relations (1.6a) and (1.6b), gives the following equivalent form of generating function (2.7a):



1 g ( f −1 (t ))







exp y f −1 (t ) C 0 xf −1 (t ) =

∞ 

L sn (x,

y)

n =0

tn n!

(2.7b)

.

Again, taking x = 0 in Eq. (2.7a) (or (2.7b)) and using definitions (1.1) and (1.28), we have L sn (0,

y ) = sn ( y ).

(2.8)

Taking y = 0 in Eq. (2.6) and using definition (1.1), we have L sn (x, 0)

  1 . = sn − D − x

(2.9)

Also, from Eq. (2.7a) (or (2.7b)), we note that

∂ ∂ ∂ x L sn (x, y ), L sn (x, y ) = − ∂y ∂x ∂x

(2.10)

which in view of Eq. (1.35), can be written as

∂ ∂ s (x, y ). L sn (x, y ) = − 1L n ∂y ∂ D− x

(2.11)

The formal solution of Eq. (2.11) along with initial condition (2.8) is given by



L sn (x,

1 y ) = exp − D − x

∂ sn ( y ), ∂y

(2.12)

or, equivalently the solution of Eq. (2.11) along with initial condition (2.9) is given by



L sn (x,

y ) = exp − y



∂ −1

∂ Dx





1 . sn − D − x

(2.13)

Further, using shift identity (2.2) in Eqs. (2.12) or (2.13), we have L sn (x,





1 . y ) = sn y − D − x

(2.14)

The operational rules (2.12) and (2.13) provide a correspondence between the Sheffer and the Laguerre–Sheffer polynomials. A simple computation shows that these operational rules can be written in the following general form:







−1 L sn mx, m( y + z) = exp − D x

and

  L sn mx, m( y + z) = exp − y 

     ∂ 1 ∂ sn m( y + z) = exp z − D − sn (my ) x ∂y ∂y

∂



 

−1

sn m z − D x

−1

∂ Dx



 = exp −( y + z)

∂ −1

∂ Dx





(2.15)



1 . sn −mD − x

(2.16)

From any of Eqs. (2.14)–(2.16), it follows that L sn





 



1 . mx, m( y + z) = sn m y + z − D − x

(2.17)

Now, making use of identity (2.2) in Eq. (2.17), we find

  L sn mx, m( y + z) = exp − z 

∂ exp − D x sn (my ) 1 ∂y ∂ D− x   ∂ 1 ∂ = exp − y −1 exp − D − sn (mz). x ∂z ∂ Dx ∂





−1

(2.18)

Also, for z = 0, Eqs. (2.15)–(2.17), become

 1 ∂ = exp − D − sn (my ), x ∂y    ∂ 1 sn −mD − L sn (mx, my ) = exp − y x −1 ∂ Dx L sn (mx, my )

(2.19) (2.20)

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S. Khan, N. Raza / J. Math. Anal. Appl. 387 (2012) 90–102

and L sn (mx, my )

   1 = sn m y − D − , x

(2.21)

respectively. For x = 0, Eq. (2.21) gives L sn (0, my )

= sn (my )

(2.22)

and for y = 0, it gives L sn (mx, 0)

  1 = sn −mD − . x

(2.23)

Eqs. (2.22) and (2.23) are the generalized form of Eqs. (2.8) and (2.9) respectively. Also, in view of Eq. (2.19), we can express Eq. (2.18) as: L sn







mx, m( y + z) = exp − y



∂

L sn (mx, mz)

1 ∂ D− x

 = exp − z

∂ 1 ∂ D− x



L sn (mx, my ).

(2.24)

We note that for H (t ) = t, generating function (2.7a), yields the generating function for the Laguerre–Appell polynomials y ) as:

L an (x,

A (t ) exp( yt )C 0 (xt ) =

∞ 

L an (x,

y)

n =0

tn n!

(2.25a)

.

Also, for f −1 (t ) = t, generating function (2.7b) yields the following equivalent form of generating function (2.25a):

1 g (t )

exp( yt )C 0 (xt ) =

∞ 

L an (x,

y)

n =0

tn n!

(2.25b)

.

Using Eqs. (1.2a) and (1.25) in the l.h.s. of Eq. (2.25a), it becomes ∞  ∞ 

A n L k (x, y )

n =0 k =0

t n+k n!k!

=

∞ 

L an (x,

y)

n =0

tn n!

(2.26)

,

which on equating the coefficients of equal powers of t, gives the series definition for L an (x, y ) as: L an (x, y ) = n!

n  A n−k L k (x, y ) . (n − k)!k!

(2.27)

k =0

3. Monomiality principle and differential equation Theorem 3.1. The Laguerre–Sheffer polynomials L sn (x, y ) are quasi-monomial under the action of the following multiplicative and derivative operators:

ˆ Ls = M



1 y − D− x −

g(D y ) g(D y )



1 f (D y )

(3.1a)

,

or









1 ˆ Ls = y − D − M H  H −1 ( D y ) + x

A  ( H −1 ( D y )) A ( H −1 ( D y ))

,

(3.1b)

and

Pˆ Ls = f ( D y ),

(3.2a)

Pˆ Ls = H −1 ( D y ),

(3.2b)

or

respectively. Proof. In view of Eqs. (1.12) and (1.18a), we have



y−

g(D y ) g(D y )



1 f (D y )





sn ( y ) = sn+1 ( y ).

S. Khan, N. Raza / J. Math. Anal. Appl. 387 (2012) 90–102

97

1 Operating exp(− D − x D y ) on both sides of the above equation, we find



−1

exp − D x D y



so that, we have



−1

exp − D x D y





g(D y )

y−





1 f (D y )

g(D y )









sn+1 ( y ) ,

    g(D y ) 1

 

   −1 1 y  sn ( y ) − exp − D x D y sn ( y ) = exp − D − x D y sn+1 ( y ) , f (D y ) g(D y ) f (D y )

1

which on using identity (2.2) in the l.h.s., becomes





1 sn ( y ) = exp − D − x Dy



1 1 y − D− exp − D − x x Dy





1 f (D y )





sn ( y )

 1 = exp − D − x D y sn+1 ( y ) . 



   g(D y ) 1

 1 − exp − D − D s ( y ) y n x g(D y ) f (D y ) (3.3)

Now, since for any arbitrary function φ( D y ) of D y , we have









1 exp − D − x D y , φ( D y ) = 0,

therefore, Eq. (3.3), becomes



−1

y − Dx −

g(D y )



(3.4)



1 f (D y )

g(D y )

−1

exp − D x D y







1 sn ( y ) = exp − D − x Dy





sn+1 ( y ) .

Finally, using operational definition (2.12) in the above equation, we get



1 y − D− x −

g(D y )





1 f (D

g(D y )



y)

L sn (x,



y ) = L sn+1 (x, y ),

(3.5)

which in view of Eq. (1.12) yields assertion (3.1a) of Theorem 3.1. Also, in view of relations (1.6a) and (1.6b), Eq. (3.1a) gives assertion (3.1b). Similarly, in view of Eqs. (1.13) and (1.19a), we have





f ( D y ) sn ( y ) = nsn−1 ( y ). 1 Operating exp(− D − x D y ) on both sides of the above equation, we find



1 exp − D − x Dy







f ( D y ) sn ( y )

 

 1 = exp − D − x D y nsn−1 ( y ) ,

which on using relation (3.4) in the l.h.s., becomes



1 f ( D y ) exp − D − x Dy







1 sn ( y ) = exp − D − x Dy



(3.6)



nsn−1 ( y ) .

Further, using operational definition (2.12) in the above equation, we find



f (D y )

L sn (x,



y ) = n L sn−1 (x, y ),

(3.7)

which in view of Eq. (1.13) yields assertion (3.2a) of Theorem 3.1. Also, in view of relation (1.6b), Eq. (3.2a) gives assertion (3.2b) of Theorem 3.1. 2 Remark 3.1. Eqs. (3.5) and (3.7) are the recurrence relations satisfied by the Laguerre–Sheffer polynomials L sn (x, y ). Remark 3.2. In view of relation (2.11), the multiplicative and derivative operators (3.1a), (3.1b) and (3.2a), (3.2b) of the Laguerre–Sheffer polynomials L sn (x, y ) can also be expressed as:

ˆ Ls = M



  g  − ∂−1

1 y − D− x −

 g −

∂ Dx  ∂ 1 ∂ D− x

f



1

−

∂ 1 ∂ D− x

or, equivalently



 −1

ˆ Ls = y − D x M and

 ˆP Ls = f −

∂ 1 ∂ D− x

H



 −1 − H

∂ −1

∂ Dx

+

,

(3.8a)

   A  H −1 − ∂−1 ∂ Dx

   , A H −1 − ∂−1

(3.8b)

∂ Dx

,

(3.9a)

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S. Khan, N. Raza / J. Math. Anal. Appl. 387 (2012) 90–102

or, equivalently

 ˆP Ls = H −1 −



∂

(3.9b)

,

1 ∂ D− x

respectively. Theorem 3.2. The Laguerre–Appell polynomials L an (x, y ) are quasi-monomial under the action of the following multiplicative and derivative operators:

A(D y )

1 ˆ La = y − D − M x +

A(D y )

(3.10a)

,

or, equivalently

g(D y )

1 ˆ La = y − D − M x −

(3.10b)

g(D y )

and

Pˆ La = D y .

(3.11)

Proof. Consider the identity

D y exp( yt )C 0 (xt ) = t exp( yt )C 0 (xt ).

(3.12)

Since A (t ) can be expressed as Taylor’s series, therefore we have

A ( D y ) exp( yt )C 0 (xt ) = A (t ) exp( yt )C 0 (xt ), which in view of generating functions (1.25) and (2.25a), becomes



A(D y )

∞ 

L n (x, y )

n =0

tn



=

n!

∞ 

L an (x,

y)

n =0

tn n!

.

Now, equating the coefficients of like powers of t in the above equation, we find





A ( D y ) L n (x, y ) = L an (x, y ), or, equivalently



(3.13)



A ( D y ) L n+1 (x, y ) = L an+1 (x, y ), which on using Eqs. (1.12) and (1.22) in the l.h.s., gives



A(D y )





1 y − D− L n (x, y ) = L an+1 (x, y ). x

(3.14)

Since the inverse of A ( D y ) exists, we have



− 1

A(D y )

A ( D y ) = 1.

(3.15)

Therefore, operating ( A ( D y ))−1 on Eq. (3.13), we find

− 1



L n (x, y ) = A ( D y )

L an (x,



y) .

(3.16)

Using Eq. (3.16) in Eq. (3.14), we find



1 A(D y ) y − D − x



− 1

A(D y )

L an (x,



y ) = L an+1 (x, y ),

(3.17)

which in view of Eq. (1.12) shows that the multiplicative operator for L an (x, y ) is given as



1 ˆ La = A ( D y ) y − D − M x



− 1

A(D y )

.

(3.18)

Using identity



 F ( D y ), y = F  ( D y ),

(3.19)

we have



− 1

A(D y ) y A(D y )

=y+

A(D y ) A(D y )

.

(3.20)

S. Khan, N. Raza / J. Math. Anal. Appl. 387 (2012) 90–102

99

Further, using Eq. (3.20) in Eq. (3.18), we get assertion (3.10a) of Theorem 3.2. Also, in view of relation (1.8), we get assertion (3.10b). Again, using identity (3.12) in Eq. (2.25a) (or (2.25b)), we get



Dy

∞ 

L an (x,

y)

n =0

tn



=

n!

∞ 

L an−1 (x,

y)

n =1

tn

(n − 1)!

,

or, equivalently ∞ 

Dy



L an (x,

 tn

y)

=

n!

n =0

∞ 

L an−1 (x,

y)

n =1

tn

(n − 1)!

.

Equating the coefficients of like powers of t in the above equation, we get

Dy



L an (x,



y ) = n L an−1 (x, y ),

which in view of Eq. (1.13), yields assertion (3.11) of Theorem 3.2.

2

Alternate proof. Since, for f (t ) = H −1 (t ) = t, the Laguerre–Sheffer polynomials L sn (x, y ) reduce to the Laguerre–Appell polynomials L an (x, y ), therefore taking f (t ) = t and H (t ) = t in Eqs. (3.1a) and (3.1b), respectively, we get assertions (3.10b) and (3.10a). Also, taking f (t ) = t and H (t ) = t in Eqs. (3.2a) and (3.2b) respectively, we get assertion (3.11). 2 Remark 3.3. In view of relation (2.11), the multiplicative and derivative operators (3.10a), (3.10b) and (3.11) of the Laguerre– Appell polynomials L an (x, y ) can also be expressed as:

ˆ La = y − D x + M −1

  A  − ∂−1 ∂ Dx

 , A − ∂−1

(3.21a)

∂ Dx

or, equivalently

ˆ La = y − D x − M −1

  g  − ∂−1 ∂ Dx

 , g − ∂−1

(3.21b)

∂ Dx

and

Pˆ La = −

∂ 1 ∂ D− x

(3.22)

,

respectively. Theorem 3.3. The Laguerre–Sheffer polynomials L sn (x, y ) satisfy the following differential equation:



xy or,

g(D y ) f (D y ) ∂ ∂2 + − ( x − y ) + D n y L sn (x, y ) = 0, ∂x g(D y ) D y f (D y ) ∂ x2

 xy

A  ( H −1 ( D y )) ∂ H  ( H −1 ( D y )) ∂2 −1 − ( x − y ) − H ( D ) + n y L sn (x, y ) = 0. ∂x Dy ∂ x2 A ( H −1 ( D y ))

(3.23a)

(3.23b)

Proof. Using Eqs. (3.1a) and (3.2a) in monomiality principle equation (1.15), we have



or,



1 y − D− x −





g(D y )



g(D y )

1 y − D− Dy x

f (D y )

f (D

f (D y ) D y f (D y )

y)

−

L sn (x,



y ) = n L sn (x, y ),

g(D y ) f (D y ) g(D y ) f (D y )

(3.24)

L sn (x,

y ) = n L sn (x, y ),

which on using relation [6, p. 455 (12)]:





1 y − D− D y = −xy x

∂ ∂2 + (x − y ) , ∂x ∂ x2

(3.25)

yields Eq. (3.23a). Again, using Eqs. (3.1b) and (3.2b) in monomiality principle equation (1.15), we get assertion (3.23b).

2

100

S. Khan, N. Raza / J. Math. Anal. Appl. 387 (2012) 90–102

Remark 3.4. In view of relation (2.11), Eqs. (3.23a) and (3.23b) can also be expressed as:

  x y− and



  g  − ∂−1 ∂ 2 ∂ Dx  g −



∂ 1 ∂ D− x

  g  − ∂−1 ∂ ∂ Dx



∂ x2

− x− y+

 g −

∂ 1 ∂ D− x



∂x



  f − ∂−1 ∂D

 x − ∂−1 f  − ∂ Dx

∂ 1 ∂ D− x

 +n

L sn (x,

y) = 0

       −1 − ∂−1  A  H −1 − ∂−1 ∂ ∂ H H ∂2 ∂ Dx ∂ Dx −1 − + − xy 2 − (x − y ) H n      L sn (x, y ) = 0, 1 ∂x ∂x − ∂−1 A H −1 − ∂−1 ∂ D− x ∂ Dx

(3.26a)

(3.26b)

∂ Dx

respectively. Remark 3.5. Since, for f (t ) = H −1 (t ) = t, the Laguerre–Sheffer polynomials L sn (x, y ) reduce to the Laguerre–Appell polynomials L an (x, y ), therefore for f (t ) = H −1 (t ) = t, we deduce the following consequence of Theorem 3.3. Corollary 3.1. The Laguerre–Appell polynomials L an (x, y ) satisfy the following differential equation



g(D y ) ∂ ∂2 ∂ xy 2 − (x − y ) + +n ∂x g(D y ) ∂ y ∂x

L an (x,

y ) = 0,

(3.27a)

y ) = 0.

(3.27b)

or, equivalently

 xy

A(D y ) ∂ ∂ ∂2 − ( x − y ) − +n ∂x A(D y ) ∂ y ∂ x2

L an (x,

Remark 3.6. In view of relation (2.11), Eqs. (3.27a) and (3.27b) can also be expressed as:

  x y−

  g  − ∂−1 ∂ 2 ∂ Dx 

g − ∂−1



∂ x2

∂ Dx

and

  x y+

  A  − ∂−1 ∂ 2 ∂ Dx 

A − ∂−1



∂ x2

∂ Dx

 − x− y+

  g  − ∂−1 ∂ ∂ Dx 

g − ∂−1



∂ Dx

 − x− y−

∂x

  A  − ∂−1 ∂ ∂ Dx 

A − ∂−1 ∂ Dx



∂x

+n

L an (x,

y) = 0

(3.28a)

y ) = 0.

(3.28b)

+n

L an (x,

4. Concluding remarks In the previous two sections, the family of Laguerre–Sheffer polynomials is introduced and its properties are derived within the context of monomiality principle and by using certain operational techniques. Further, to bolster the contention of using operational techniques in the theory of special polynomials, we derive the integral representation for the Laguerre–Sheffer polynomials. Dattoli et al. [7] have shown that the integral transform method associated with operational techniques provides an effective way to obtain the practical solutions of families of partial differential equations. We recall the following wave-type equation [7, p. 661 (1)]:

∂ ∂ ∂ F (x, y ) = − x F (x, y ), ∂y ∂x ∂x F (x, 0) = g (x),

(4.1) (4.2)

whose formal solution can be written as [7, p. 661 (2)]:





∂ ∂ F (x, y ) = exp − y x ∂x ∂x

g (x).

(4.3)

The solution of Eq. (4.1) is specified by the following integral representation [5, p. 226 (51)]:

 +∞ F (x, y ) = exp

x

exp(−τ )C 0

y 0



x y



τ g (− y τ ) dτ ( y = 0).

(4.4)

S. Khan, N. Raza / J. Math. Anal. Appl. 387 (2012) 90–102

101

Now, since L sn (x, y ) satisfy Eqs. (2.9) and (2.10), therefore, in view of Eqs. (4.1) and (4.2), we can use integral represen1 tation (4.4), with F (x, y ) = L sn (x, y ) and F (x, 0) = L sn (x, 0) = g (x) = sn (− D − x ) to get

 +∞ L sn (x, y ) = exp

x



exp(−τ )C 0

y



x y





τ sn D −yτ1 dτ ( y = 0).

(4.5)

0

Again, since 1 −1 D− yτ = τ D y ,

(4.6)

therefore, Eq. (4.5) becomes

 +∞ L sn (x,

y ) = exp

x



exp(−τ )C 0

y



x y





τ sn τ D −y 1 dτ ( y = 0),

(4.7)

0

which is the integral representation for expressed as:

L sn (x,

 +∞ L sn (x,

y ) = exp

x



exp(−τ )C 0

y

y ). Also, in view of Eq. (2.23), the above integral representation can be



x y

τ y , 0) dτ ( y = 0).

τ

L sn (−

(4.8)

0

Next, we use the fact that the inverse of the operator A ( D y ) exists to introduce the complementary Laguerre–Appell polynomials L an (x, y ) in the following manner:

L an (x,



− 1

y) = A( D y )

L n (x, y ).

(4.9)

Now, using the same lines of proof of Theorem 3.2, we observe that L an (x, y ) are quasi-monomial under the action of the following multiplicative and derivative operators: 1 ˆ La = y − D − M x − 1 ˆ La = y − D − M x +

A(D y ) A(D y )

(4.10a)

,

g(D y )

(4.10b)

g(D y )

and

Pˆ La = D y ,

(4.11)

respectively. Also, the complementary Laguerre–Appell polynomials L an (x, y ) satisfy the following differential equation:

 xy

g(D y ) ∂ ∂ ∂2 − (x − y ) − +n 2 ∂x g(D y ) ∂ y ∂x



y ) = 0,

(4.12a)



y ) = 0.

(4.12b)

L an (x,

or, equivalently

 xy

A(D y ) ∂ ∂ ∂2 − (x − y ) + +n 2 ∂x A(D y ) ∂ y ∂x

L an (x,

In this paper, the family of Laguerre–Sheffer polynomials L sn (x, y ) are introduced by considering the generating function (1.1) of Sheffer polynomials sn (x) and the multiplicative operator (1.22) of the base polynomials i.e. of 2VLP L n (x, y ), thus establishing a correspondence between the families of Sheffer and the Laguerre–Sheffer polynomials. In a forthcoming investigation, we explore the possibility of establishing a connection between L sn (x, y ) and L n (x, y ). We remark that such a connection between Laguerre–Appell polynomials L an (x, y ) and L n (x, y ) is given in Eq. (3.13). Appendix A We remark that by making use of the results derived in the preceding sections, we can establish the corresponding results for the Laguerre–Sheffer polynomials by taking A (t ) and H (t ) (or f (t ) and g (t )) of the corresponding Sheffer polynomials given in Table A.1.

102

S. Khan, N. Raza / J. Math. Anal. Appl. 387 (2012) 90–102

Table A.1 List of known Sheffer polynomials. S. No.

A (t ); H (t )

I.

m e −t ;

II. III.

g (t ); f (t )

νt

e

(1 − t )−α −1 ; t 1−t βt

t t −1

t ; ln( 11+ −t )

( νt )m

Generating functions

t

exp(ν xt − t ) = m

; ν

(1 − t )−α −1 ;

1

t t −1

(1−t )α +1

t 2 ; e −1 et −1 et +1

t 1−t

tn n=0 H n,m,ν (x) n!

xt exp ( t − )= 1

t x ( 11+ −t ) =

Polynomials

∞

∞

∞

n=0

P n (x) tn!

exp(β t + x(1 − e )) =

∞

e ; 1−e

V.

e −t ; ln(1 + at )

exp(a(e − 1)); a(e − 1)

e −t (1 + t )x

VI.

(1 + (1 + t )λ )−μ ; ln(1 + t )

(1 + e λt )μ ; et

(1 + (1 + t )λ )−μ (1 + t )x

t

t

−1

t

a

t ; ln(1+t )

VIII.

2 ; 2+t √1

IX. X.

ln(1 + t )

ln(1 + t )

1+t 2

; arctan(t ) 1 2

(1 − 4t )− ( √−4t (1+ 1−4t )2

√2 )a−1 ; 1+ 1−4t

t ; et −1

et − 1

1 (1 + et ); 2

et − 1

sec t; tan t 1+t (1−t )a

;

1 4

t 2 − 14 ( 11+ −t )

∞

Generalized Laguerre polynomials (α ) n! L n (x) [1,18] Pidduck polynomials P n (x) [3,14]

(β)

n=0 an ∞ tn n=0 c n (x; a) n!

n

(β)

(x) tn!

Acturial polynomials an (x) [3] Poisson–Charlier polynomials cn (x; a) [13,15,22] Peters polynomials sn (x; λ, μ) [3]

tn n=0 sn (x; λ, ) n!  n t (1 + t )x = n∞=0 bn (x) tn! ln(1+t )

=

VII.

=

L n (x)t n

n

n=0

(1 − t )−β ; ln(1 − t )

IV.

t

Generalized Hermite polynomials H n,m,ν (x) [16]

(α )

μ

2 (1 + t )x 2+t 1 √ 1+t 2

=

∞

tn n=0 rn (x) n!

exp(x arctan(t )) = 1 2

(1 − 4t )− (

Bernoulli polynomials of the second kind bn (x) [15] Related polynomials rn (x) [15]

∞

n=0

n

R n (x) tn!

√2 )a−1 1+ 1−4t  −4xt × exp( √ ) = n∞=0 R n (a, x)t n (1+ 1−4t )2

Hahn polynomials R n (x) [2] Shively’s pseudo-Laguerre polynomials R n (a, x) [18]

References [1] L.C. Andrews, Special Functions for Engineers and Applied Mathematicians, Macmillan, New York, 1985. [2] 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. [3] R.P. Boas, R.C. Buck, Polynomial Expansions of Analytic Functions, Springer-Verlag, Berlin, Gottingen, Heidelberg, 1958. [4] G. Dattoli, Hermite–Bessel and Laguerre–Bessel functions: A by-product of the monomiality principle. Advanced special functions and applications, in: Proc. Melfi Sch. Adv. Top. Math. Phys., vol. 1, Melfi, 1999, Aracne, Rome, 2000, pp. 147–164. [5] G. Dattoli, B. Germano, P.E. Ricci, Comments on monomiality, ordinary polynomials and associated bi-orthogonal functions, Appl. Math. Comput. 154 (2004) 219–227. [6] G. Dattoli, Subuhi Khan, Monomiality, Lie algebras and Laguerre polynomials, J. Appl. Funct. Anal. 1 (4) (2006) 453–460. [7] G. Dattoli, M.R. Martinelli, P.E. Ricci, On new families of integral transforms for the solution of partial differential equations, Integral Transforms Spec. Funct. 16 (8) (2005) 661–667. [8] G. Dattoli, M. Migliorati, H.M. Srivastava, Sheffer polynomials, monomiality principle, algebraic methods and the theory of classical polynomials, Math. Comput. Modelling 45 (2007) 1033–1041. [9] G. Dattoli, P.L. Ottaviani, A. Torre, L. Vázquez, Evolution operator equations: integration with algebraic and finite difference methods. Applications to physical problems in classical and quantum mechanics and quantum field theory, Riv. Nuovo Cimento Soc. Ital. Fis. (4) 20 (2) (1997) 1–133. [10] G. Dattoli, H.M. Srivastava, K. Zhukovsky, A new family of integral transforms and their applications, Integral Transforms Spec. Funct. 17 (1) (2006) 31–36. [11] G. Dattoli, A. Torre, Operational methods and two variable Laguerre polynomials, Atti Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur. 132 (1998) 1–7. [12] G. Dattoli, A. Torre, Exponential operators, quasi-monomials and generalized polynomials, Radiation Phys. Chem. 57 (2000) 21–26. [13] A. Erdélyi, W. Magnus, F. Oberhettinger, F.G. Tricomi, Higher Transcendental Function. The Bateman Manuscript Project, vol. II, McGraw–Hill, New York, Toronto, London, 1953. [14] A. Erdélyi, W. Magnus, F. Oberhettinger, F.G. Tricomi, Higher Transcendental Function. The Bateman Manuscript Project, vol. III, McGraw–Hill, New York, Toronto, London, 1955. [15] C. Jordan, Calculus of Finite Differences, Chelsea, Bronx, New York, 1965. [16] M. Lahiri, On a generalization of Hermite polynomials, Proc. Amer. Math. Soc. 27 (1971) 117–121. [17] 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, J. Phys. Conf. Ser. 30 (2006) 86–97. [18] E.D. Rainville, Special Functions, Macmillan, New York, 1960; Reprinted by Chelsea Publ. Co., Bronx, New York, 1971. [19] S. Roman, The Umbral Calculus, Academic Press, New York, 1984. [20] H.M. Srivastava, H.L. Manocha, A Treatise on Generating Functions, Ellis Horwood Limited, New York, 1984. [21] J.F. Steffensen, The poweroid, an extension of the mathematical notion of power, Acta Math. 73 (1941) 333–366. [22] G. Szegö, Orthogonal Polynomials, Amer. Math. Soc., Providence, RI, 1978. [23] A. Wrulich, Beam LifeTime in Storage Rings, CERN Accelerator School, 1992.