Two-parameter Srivastava polynomials and several series identities

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Abstract

In the present paper, we introduce two-parameter Srivastava polynomials in one, two and three variables by inserting new indices, where in the special cases they reduce to (among others) Laguerre, Jacobi, Bessel and Lagrange polynomials. These polynomials include the family of polynomials which were introduced and/or investigated in (Srivastava in Indian J. Math. 14:1-6, 1972; González et al. in Math. Comput. Model. 34:133-175, 2001; Altın et al. in Integral Transforms Spec. Funct. 17(5):315-320, 2006; Srivastava et al. in Integral Transforms Spec. Funct. 21(12):885-896, 2010; Kaanoglu and Özarslan in Math. Comput. Model. 54:625-631, 2011). We prove several two-sided linear generating relations and obtain various series identities for these polynomials. Furthermore, we exhibit some illustrative consequences of the main results for some well-known special polynomials which are contained by the two-parameter Srivastava polynomials.

MSC:33C45.

Keywords

Complex Number Suitable Choice Jacobi Polynomial Laguerre Polynomial Double Sequence 

1 Introduction

Let { A n , k } n , k = 0 Open image in new window be a bounded double sequence of real or complex numbers, let [ a ] Open image in new window denote the greatest integer in a R Open image in new window, and let ( λ ) ν , ( λ ) 0 1 Open image in new window, denote the Pochhammer symbol defined by
( λ ) ν : = Γ ( λ + ν ) Γ ( λ ) Open image in new window
by means of familiar gamma functions. In 1972, Srivastava [1] introduced the following family of polynomials:
S n N ( z ) : = k = 0 [ n N ] ( n ) N k k ! A n , k z k ( n N 0 = N { 0 } ; N N ) , Open image in new window
(1)

where ℕ is the set of positive integers.

Afterward, González et al. [2] extended the Srivastava polynomials S n N ( z ) Open image in new window as follows:
S n , m N ( z ) : = k = 0 [ n N ] ( n ) N k k ! A n + m , k z k ( m , n N 0 ; N N ) Open image in new window
(2)

and investigated their properties extensively. Motivated essentially by the definitions (1) and (2), scientists investigated and studied various classes of Srivastava polynomials in one and more variables.

In [3], the following family of bivariate polynomials was introduced:
S n m , N ( x , y ) : = k = 0 [ n N ] A m + n , k x n N k ( n N k ) ! y k k ! ( n , m N 0 , N N ) , Open image in new window

and it was shown that the polynomials S n m , N ( x , y ) Open image in new window include many well-known polynomials such as Lagrange-Hermite polynomials, Lagrange polynomials and Hermite-Kampé de Feriét polynomials.

In [4], Srivastava et al. introduced the three-variable polynomials
where { A m , n , k } Open image in new window is a triple sequence of complex numbers. Suitable choices of { A m , n , k } Open image in new window in equation (3) give a three-variable version of well-known polynomials (see also [5]). Recently, in [6], the multivariable extension of the Srivastava polynomials in r-variable was introduced

where { A m , k r 1 , k 1 , k 2 , , k r 2 } Open image in new window is a sequence of complex numbers.

In this paper we introduce the two-parameter Srivastava polynomials in one and more variables by inserting new indices. These polynomials include the family of polynomials which were introduced and/or investigated in [1, 2, 3, 4, 6, 7] and [8]. We prove several two-sided linear generating relations and obtain various series identities for these polynomials. Furthermore, we exhibit some illustrative consequences of the main results for some well-known special polynomials which are contained by the two-parameter Srivastava polynomials.

2 Two-parameter one-variable Srivastava polynomials

In this section we introduce the following family of two-parameter one-variable polynomials:
S n m 1 , m 2 ( x ) : = k = 0 n ( n ) k k ! A m 1 + m 2 + n , m 2 + k x k ( m 1 , m 2 , n , k N 0 ) , Open image in new window
(5)

where { A n , k } Open image in new window is a bounded double sequence of real or complex numbers. Note that appropriate choices of the sequence A n , k Open image in new window give one-variable versions of the well-known polynomials.

Remark 2.1 Choosing A m , n = ( α m ) n Open image in new window ( m , n N 0 Open image in new window) in (5), we get
S n m 1 , m 2 ( 1 x ) = ( 1 ) m 2 ( α + m 1 + n + 1 ) m 2 n ! ( x ) n L n ( α + m 1 ) ( x ) , Open image in new window
where L n ( α ) ( x ) Open image in new window are the classical Laguerre polynomials given by
L n ( α ) ( x ) = ( x ) n n ! 2 F 0 ( n , α n ; ; 1 x ) . Open image in new window

Remark 2.2

Setting
A m , n = ( α + β + 1 ) 2 m ( β m ) n ( α + β + 1 ) m ( α β 2 m ) n ( m , n N 0 ) Open image in new window
in (5), we obtain

where P n ( α , β ) ( x ) Open image in new window are the classical Jacobi polynomials.

Remark 2.3 If we set A m , n = ( α + m 1 ) n Open image in new window ( m , n N 0 Open image in new window) in (5), then we get
S n m 1 , m 2 ( x β ) = ( α + m 1 + m 2 + n 1 ) m 2 y n ( x , α + m 1 + 2 m 2 , β ) ( β 0 ) , Open image in new window
where y n ( x , α , β ) Open image in new window are the classical Bessel polynomials given by
y n ( x , α , β ) = 2 F 0 ( n , α + n 1 ; ; x β ) . Open image in new window
Theorem 2.4 Let { f ( n ) } n = 0 Open image in new window be a bounded sequence of complex numbers. Then

provided each member of the series identity (6) exists.

Proof Let the left-hand side of (6) be denoted by Ψ ( x ) Open image in new window. Then, using the definition of S n m 1 , m 2 ( x ) Open image in new window on the left-hand side of (6), we have
Ψ ( x ) = m 1 , m 2 , n = 0 f ( n + m 1 + m 2 ) k = 0 n ( n ) k k ! A m 1 + m 2 + n , m 2 + k x k w 1 m 1 m 1 ! w 2 m 2 m 2 ! t n n ! = m 1 , m 2 , n = 0 f ( n + m 1 + m 2 ) k = 0 n 1 k ! A m 1 + m 2 + n , m 2 + k ( x ) k w 1 m 1 m 1 ! w 2 m 2 m 2 ! t n ( n k ) ! = m 1 , m 2 , n , k = 0 f ( n + m 1 + m 2 + k ) ( x t ) k k ! A m 1 + m 2 + n + k , m 2 + k w 1 m 1 m 1 ! w 2 m 2 m 2 ! t n n ! . Open image in new window
Let m 1 m 1 n Open image in new window,
Ψ ( x ) = m 1 , m 2 , k = 0 f ( m 1 + m 2 + k ) ( x t ) k k ! A m 1 + m 2 + k , m 2 + k ( n = 0 m 1 w 1 m 1 n t n ( m 1 n ) ! n ! ) w 2 m 2 m 2 ! = m 1 , m 2 , k = 0 f ( m 1 + m 2 + k ) ( x t ) k k ! A m 1 + m 2 + k , m 2 + k ( n = 0 m 1 ( m 1 n ) w 1 m 1 n t n m 1 ! ) w 2 m 2 m 2 ! = m 1 , m 2 , k = 0 f ( m 1 + m 2 + k ) A m 1 + m 2 + k , m 2 + k ( w 1 + t ) m 1 m 1 ! w 2 m 2 m 2 ! ( x t ) k k ! . Open image in new window
Let m 2 m 2 k Open image in new window,
Ψ ( x ) = m 1 , m 2 = 0 f ( m 1 + m 2 ) A m 1 + m 2 , m 2 ( w 1 + t ) m 1 m 1 ! k = 0 m 2 w 2 m 2 k ( x t ) k ( m 2 k ) ! k ! = m 1 , m 2 = 0 f ( m 1 + m 2 ) A m 1 + m 2 , m 2 ( w 1 + t ) m 1 m 1 ! ( w 2 + ( x t ) ) m 2 m 2 ! . Open image in new window

 □

Remark 2.5 Choosing A m , n = ( α m ) n Open image in new window ( m , n N 0 Open image in new window) and x 1 x Open image in new window, then by Theorem 2.4, we get
If we set w 2 = t x Open image in new window and f = 1 Open image in new window into the above equation, then we have

Remark 2.6

Setting
A m , n = ( α + β + 1 ) 2 m ( β m ) n ( α + β + 1 ) m ( α β 2 m ) n ( m , n N 0 ) Open image in new window
and x 2 1 + x Open image in new window in equation (6), we have
Remark 2.7 If we set A m , n = ( α + m 1 ) n Open image in new window ( m , n N 0 Open image in new window) and x x β Open image in new window in (6), then we can write
Remark 2.8 If we consider Remarks 2.5 and 2.7, we get the following relation between Laguerre polynomials L n ( α ) ( x ) Open image in new window and Bessel polynomials y n ( x , α , β ) Open image in new window:

3 Two-parameter two-variable Srivastava polynomials

In this section we introduce the following two-parameter family of bivariate polynomials:
S n m 1 , m 2 ( x , y ) : = k = 0 n A m 1 + m 2 + n , m 2 + k x k k ! y n k ( n k ) ! ( m 1 , m 2 , n , k N 0 ) , Open image in new window
(7)

where { A n , k } Open image in new window is a bounded double sequence of real or complex numbers. Note that in the particular case these polynomials include the Lagrange polynomials.

Remark 3.1 Choosing A m , n = ( α ) m n ( β ) n Open image in new window ( m , n N 0 Open image in new window) in (7), we have
S n m 1 , m 2 ( x , y ) = ( α ) m 1 ( β ) m 2 g n ( α + m 1 , β + m 2 ) ( x , y ) , Open image in new window
where g n ( α , β ) ( x , y ) Open image in new window are the Lagrange polynomials given by
g n ( α , β ) ( x , y ) = k = 0 n ( α ) n k ( β ) k ( n k ) ! k ! x k y n k . Open image in new window

Using similar techniques as in the proof of Theorem 2.4, we get the following theorem.

Theorem 3.2 Let { f ( n ) } n = 0 Open image in new window be a bounded sequence of complex numbers. Then

provided each member of the series identity (8) exists.

Remark 3.3 If we set A m , n = ( α ) m n ( β ) n Open image in new window ( m , n N 0 Open image in new window) in (8), we have
Furthermore, since we have the relation between P n ( α , β ) ( x , y ) Open image in new window Jacobi polynomials and Lagrange polynomials [9] as
g n ( α , β ) ( x , y ) = ( y x ) n P n ( α n , β n ) ( x + y x y ) , Open image in new window
we get the following generating relation for Jacobi polynomials P n ( α , β ) ( x , y ) Open image in new window:

4 Two-parameter three-variable Srivastava polynomials

In this section we define two-parameter three-variable Srivastava polynomials as follows:

where { A n , k , l } n , k = 0 Open image in new window is a bounded triple sequence of real or complex numbers.

Using similar techniques as in the proof of Theorem 2.4, we get the following theorem.

Theorem 4.1 Let { f ( n ) } n = 0 Open image in new window be a bounded sequence of complex numbers. Then

provided each member of the series identity (10) exists.

Theorem 4.2 Let { f ( n ) } n = 0 Open image in new window be a bounded sequence of complex numbers, and let S n m 1 , m 2 , M ( x , y , z ) Open image in new window be defined by (9). Suppose also that two-parameter two-variable polynomials P m 1 , m 2 M ( x , y ) Open image in new window are defined by
P m 1 , m 2 M ( x , y ) = l = 0 [ m 2 M ] A m 1 + m 2 , m 2 , l x m 2 M l ( m 2 M l ) ! y l l ! . Open image in new window
(11)
Then the family of two-sided linear generating relations holds true between the two-parameter three-variable Srivastava polynomials S n m 1 , m 2 , M ( x , y , z ) Open image in new window and P m 1 , m 2 M ( x , y ) Open image in new window:

Suitable choices of A n , k , l Open image in new window in equations (9) and (11) give some known polynomials.

Remark 4.3 Choosing M = 2 Open image in new window and A m , n , k = ( α ) m n ( γ ) n 2 k ( β ) k Open image in new window ( m , n N 0 Open image in new window) in (9), we get
S n m 1 , m 2 , 2 ( x , y , z ) = ( α ) m 1 ( γ ) m 2 u n ( α + m 1 , β , γ + m 2 ) ( z , x , y ) , Open image in new window
where u n ( α , β , γ ) ( x , y , z ) Open image in new window is the polynomial given by
u n ( α , β , γ ) ( x , y , z ) = k = 0 n l = 0 [ k 2 ] ( β ) l ( γ ) k 2 l ( α ) n k y l l ! x n k ( n k ) ! z k 2 l ( k 2 l ) ! . Open image in new window
Now, by setting M = 1 Open image in new window and A m , n , k = ( α ) k ( β ) n k ( γ ) m n Open image in new window ( m , n N 0 Open image in new window) in the definition (9), we obtain
S n m 1 , m 2 , 1 ( x , y , z ) = ( γ ) m 1 ( β ) m 2 g n ( α , β + m 2 , γ + m 1 ) ( x , y , z ) , Open image in new window
where g n ( α , β , γ ) ( x , y , z ) Open image in new window are the Lagrange polynomials given by
g n ( α , β , γ ) ( x , y , z ) = k = 0 n l = 0 k ( α ) l ( β ) k l ( γ ) n k x l l ! y k l ( k l ) ! z n k ( n k ) ! . Open image in new window
Remark 4.4 If we set M = 2 Open image in new window and A m , n , k = ( α ) m n ( γ ) n 2 k ( β ) k Open image in new window ( m , n N 0 Open image in new window) in (11), then
P m 1 , m 2 2 ( x , y ) = ( α ) m 1 h m 2 ( γ , β ) ( x , y ) , Open image in new window
where h m 2 ( γ , β ) ( x , y ) Open image in new window denotes the Lagrange-Hermite polynomials given explicitly by
h m 2 ( γ , β ) ( x , y ) = l = 0 [ m 2 2 ] ( γ ) m 2 2 l ( β ) l x m 2 2 l ( m 2 2 l ) ! y l l ! . Open image in new window
Furthermore, choosing M = 1 Open image in new window and A m , n , k = ( α ) k ( β ) n k ( γ ) m n Open image in new window ( m , n N 0 Open image in new window) in the definition (11), we have
P m 1 , m 2 1 ( x , y ) = ( γ ) m 1 g m 2 ( β , α ) ( x , y ) , Open image in new window
where g m 2 ( α , β ) ( x , y ) Open image in new window are the Lagrange polynomials given by
g m 2 ( α , β ) ( x , y ) = l = 0 m 2 ( α ) m 2 l ( β ) l x m 2 l ( m 2 l ) ! y l l ! . Open image in new window
Remark 4.5 If we set w 1 z t Open image in new window and w 2 y t Open image in new window in Theorem 4.1, then we get
Now, if we set f = 1 Open image in new window, M = 2 Open image in new window and A m , n , k = ( α ) m n ( γ ) n 2 k ( β ) k Open image in new window ( m , n N 0 Open image in new window) in (13), then
m 1 , m 2 , n = 0 ( α ) m 1 ( γ ) m 2 u n ( α + m 1 , β , γ + m 2 ) ( z , x , y ) ( z t ) m 1 m 1 ! ( y t ) m 2 m 2 ! t n = ( 1 x t 2 ) β . Open image in new window
Furthermore, if we set M = 2 Open image in new window and A m , n , k = ( α ) m n ( γ ) n 2 k ( β ) k Open image in new window ( m , n N 0 Open image in new window) in (12), then

Notes

Acknowledgements

Dedicated to Professor Hari M Srivastava.

We would like to thank the referees for their valuable comments.

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© Kaanoglu and Özarslan; licensee Springer. 2013

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Authors and Affiliations

  1. 1.Faculty of EngineeringCyprus International UniversityNicosiaTurkey
  2. 2.Department of Mathematics, Faculty of Arts and SciencesEastern Mediterranean UniversityGazimagusaTurkey

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