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# Two-parameter Srivastava polynomials and several series identities

- Cem Kaanoglu
^{1}Email author and - Mehmet Ali Özarslan
^{2}

**2013**:81

https://doi.org/10.1186/1687-1847-2013-81

© Kaanoglu and Özarslan; licensee Springer. 2013

**Received: **12 December 2012

**Accepted: **6 March 2013

**Published: **29 March 2013

## 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

where ℕ is the set of positive integers.

*et al.*[2] extended the Srivastava polynomials ${S}_{n}^{N}(z)$ as follows:

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.

and it was shown that the polynomials ${S}_{n}^{m,N}(x,y)$ include many well-known polynomials such as Lagrange-Hermite polynomials, Lagrange polynomials and Hermite-Kampé de Feriét polynomials.

*r*-variable was introduced

where $\{{A}_{m,{k}_{r-1},{k}_{1},{k}_{2},\dots ,{k}_{r-2}}\}$ 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 [1–4, 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

where $\{{A}_{n,k}\}$ is a bounded double sequence of real or complex numbers. Note that appropriate choices of the sequence ${A}_{n,k}$ give one-variable versions of the well-known polynomials.

**Remark 2.1**Choosing ${A}_{m,n}={(-\alpha -m)}_{n}$ ($m,n\in {\mathbb{N}}_{0}$) in (5), we get

**Remark 2.2**

where ${P}_{n}^{(\alpha ,\beta )}(x)$ are the classical Jacobi polynomials.

**Remark 2.3**If we set ${A}_{m,n}={(\alpha +m-1)}_{n}$ ($m,n\in {\mathbb{N}}_{0}$) in (5), then we get

**Theorem 2.4**

*Let*${\{f(n)\}}_{n=0}^{\mathrm{\infty}}$

*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 $\mathrm{\Psi}(x)$. Then, using the definition of ${S}_{n}^{{m}_{1},{m}_{2}}(x)$ on the left-hand side of (6), we have

□

**Remark 2.5**Choosing ${A}_{m,n}={(-\alpha -m)}_{n}$ ($m,n\in {\mathbb{N}}_{0}$) and $x\to -\frac{1}{x}$, then by Theorem 2.4, we get

**Remark 2.6**

**Remark 2.7**If we set ${A}_{m,n}={(\alpha +m-1)}_{n}$ ($m,n\in {\mathbb{N}}_{0}$) and $x\to -\frac{x}{\beta}$ in (6), then we can write

## 3 Two-parameter two-variable Srivastava polynomials

where $\{{A}_{n,k}\}$ 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 ${A}_{m,n}={(\alpha )}_{m-n}{(\beta )}_{n}$ ($m,n\in {\mathbb{N}}_{0}$) in (7), we have

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

**Theorem 3.2**

*Let*${\{f(n)\}}_{n=0}^{\mathrm{\infty}}$

*be a bounded sequence of complex numbers*.

*Then*

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

**Remark 3.3**If we set ${A}_{m,n}={(\alpha )}_{m-n}{(\beta )}_{n}$ ($m,n\in {\mathbb{N}}_{0}$) in (8), we have

## 4 Two-parameter three-variable Srivastava polynomials

where ${\{{A}_{n,k,l}\}}_{n,k=0}^{\mathrm{\infty}}$ 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*${\{f(n)\}}_{n=0}^{\mathrm{\infty}}$

*be a bounded sequence of complex numbers*.

*Then*

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

**Theorem 4.2**

*Let*${\{f(n)\}}_{n=0}^{\mathrm{\infty}}$

*be a bounded sequence of complex numbers*,

*and let*${S}_{n}^{{m}_{1},{m}_{2},M}(x,y,z)$

*be defined by*(9).

*Suppose also that two*-

*parameter two*-

*variable polynomials*${P}_{{m}_{1},{m}_{2}}^{M}(x,y)$

*are defined by*

*Then the family of two*-

*sided linear generating relations holds true between the two*-

*parameter three*-

*variable Srivastava polynomials*${S}_{n}^{{m}_{1},{m}_{2},M}(x,y,z)$

*and*${P}_{{m}_{1},{m}_{2}}^{M}(x,y)$:

Suitable choices of ${A}_{n,k,l}$ in equations (9) and (11) give some known polynomials.

**Remark 4.3**Choosing $M=2$ and ${A}_{m,n,k}={(\alpha )}_{m-n}{(\gamma )}_{n-2k}{(\beta )}_{k}$ ($m,n\in {\mathbb{N}}_{0}$) in (9), we get

**Remark 4.4**If we set $M=2$ and ${A}_{m,n,k}={(\alpha )}_{m-n}{(\gamma )}_{n-2k}{(\beta )}_{k}$ ($m,n\in {\mathbb{N}}_{0}$) in (11), then

## Declarations

### Acknowledgements

Dedicated to Professor Hari M Srivastava.

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

## Authors’ Affiliations

## References

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## Copyright

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