A new summation identity for the Srivastava-Singhal polynomials
type: Item
instance of: scholarly article
title: A new summation identity for the Srivastava-Singhal polynomials
zbMATH Open document ID: 1076.33005
published in: Journal of Mathematical Analysis and Applications
publication date: 2004-10-12T00:00:00Z
review text: The Srivastava-Singhal polynomials \(G_n^{(\alpha)}(x,\,r,\,\beta,\,\kappa)\) can be defined by the following Rodrigues-type formula: \[ G_n^{(\alpha)}(x,\,r,\,\beta,\,\kappa):=\frac{x^{-\alpha-\kappa\,n}}{n!}\exp(\beta\,x^r) \Big(x^{\kappa+1}\frac{d}{dx}\Big)^n\;\big\{x^{\alpha}\,\exp(-\beta\,x^{r})\big\}. \] These polynomials are represented explicitly by \[ G_n^{(\alpha)}(x,\,r,\,\beta,\,\kappa)= \frac{\kappa^n}{n!}\,\sum_{k=0}^n\frac{(-\beta\,x^{r})^{k}}{k!}\Delta_{\alpha,r}^{k}\Big(\frac{\alpha} {\kappa}\Big)_n\,, \] where \[ \Delta_{\alpha,r}^{k}f(\alpha):=\sum_{j=0}^{k}(-1)^{k-j}\Big({k \atop j}\Big)\,f(\alpha+rj)\,. \] The Srivastava-Singhal polynomials \(G_n^{(\alpha)}(x,\,r,\,\beta,\,\kappa)\) generalize the classical Hermite polynomials \(H_n(x)\) and the Laguerre polynomials \(L_n^{(\alpha)}(x)\), for appropriate choice of the parameters involved. In particular, we have \[ L_n^{(\alpha)}(x)=G_n^{(\alpha+1)}(x,1,1,1). \] The author proves in this paper the following summation formula: \[ \begin{multlined} \sum_{k=0}^{p}{\mathcal{P}}_m(k)G_{k}^{(\alpha)}(x,\,r,\,\beta,\,1)G_{p-k}^{(-\alpha-n+1)} (x,\,r,\,-\beta,\,1)\\=(-1)^{n-1}{\mathcal{P}}_m(-\alpha){\delta}_{n,\,p+1}+\frac{{\mathcal P}_m^{(m)}(0)}{m!}(-\beta r\, x^{r})^m{\delta}_{n,\,(r+1)m},\end{multlined} \] where \({\mathcal{P}}_m(x)\) is an arbitrary polynomial of degree \(m\) in \(x\) and \(m,\,n,\,p,\,r,\,\in \mathbb{N}_{0};\;\;p+1\geq n\geq (r+1)m\). As usual, \(\delta_{n,\,m}\) denotes the Kronecker symbol. Formula (1) generalizes a corresponding result of \textit{H. Bavinck} [J. Phys. A Math. Gen. 29, 277--279 (1996)] for the Laguerre polynomials \(L_n^{(\alpha)}(x)\).
reviewed by: Stamatis Koumandos
Mathematics Subject Classification ID: 33C45
Mathematics Subject Classification ID: 33C47
Mathematics Subject Classification ID: 33C50
zbMATH DE Number: 2107275
zbMATH Keywords: Laguerre polynomials
zbMATH Keywords: Hermite polynomials
zbMATH Keywords: Srivastava-Singhal polynomials
zbMATH Keywords: summation formula
zbMATH Keywords: biorthogonal polynomials
author: Kung-Yu Chen
MaRDI profile type: Publication
full work available at URL: j.jmaa.2004.05.043
OpenAlex ID: W2000567842
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DOI: 10.1016/J.JMAA.2004.05.043
description: scientific article; zbMATH DE number 2107275