, 2013:81

# Two-parameter Srivastava polynomials and several series identities

Open Access
Research
Part of the following topical collections:
1. Proceedings of the International Congress in Honour of Professor Hari M. Srivastava

## 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 be a bounded double sequence of real or complex numbers, let denote the greatest integer in , and let , denote the Pochhammer symbol defined by
by means of familiar gamma functions. In 1972, Srivastava [1] introduced the following family of polynomials:
(1)

where ℕ is the set of positive integers.

Afterward, González et al. [2] extended the Srivastava polynomials as follows:
(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:

and it was shown that the polynomials 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 is a triple sequence of complex numbers. Suitable choices of 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 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:
(5)

where is a bounded double sequence of real or complex numbers. Note that appropriate choices of the sequence give one-variable versions of the well-known polynomials.

Remark 2.1 Choosing () in (5), we get
where are the classical Laguerre polynomials given by

Remark 2.2

Setting
in (5), we obtain

where are the classical Jacobi polynomials.

Remark 2.3 If we set () in (5), then we get
where are the classical Bessel polynomials given by
Theorem 2.4 Let 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 . Then, using the definition of on the left-hand side of (6), we have
Let ,
Let ,

□

Remark 2.5 Choosing () and , then by Theorem 2.4, we get
If we set and into the above equation, then we have

Remark 2.6

Setting
and in equation (6), we have
Remark 2.7 If we set () and in (6), then we can write
If we set and , then
Remark 2.8 If we consider Remarks 2.5 and 2.7, we get the following relation between Laguerre polynomials and Bessel polynomials :

## 3 Two-parameter two-variable Srivastava polynomials

In this section we introduce the following two-parameter family of bivariate polynomials:
(7)

where 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 () in (7), we have
where are the Lagrange polynomials given by

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

Theorem 3.2 Let be a bounded sequence of complex numbers. Then

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

Remark 3.3 If we set () in (8), we have
Choosing gives
Furthermore, since we have the relation between Jacobi polynomials and Lagrange polynomials [9] as
we get the following generating relation for Jacobi polynomials :

## 4 Two-parameter three-variable Srivastava polynomials

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

where 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 be a bounded sequence of complex numbers. Then

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

Theorem 4.2 Let be a bounded sequence of complex numbers, and let be defined by (9). Suppose also that two-parameter two-variable polynomials are defined by
(11)
Then the family of two-sided linear generating relations holds true between the two-parameter three-variable Srivastava polynomials and :

Suitable choices of in equations (9) and (11) give some known polynomials.

Remark 4.3 Choosing and () in (9), we get
where is the polynomial given by
Now, by setting and () in the definition (9), we obtain
where are the Lagrange polynomials given by
Remark 4.4 If we set and () in (11), then
where denotes the Lagrange-Hermite polynomials given explicitly by
Furthermore, choosing and () in the definition (11), we have
where are the Lagrange polynomials given by
Remark 4.5 If we set and in Theorem 4.1, then we get
Now, if we set , and () in (13), then
Furthermore, if we set and () in (12), then

## Notes

### Acknowledgements

Dedicated to Professor Hari M Srivastava.

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

### References

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2. 2.
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