A new summation identity for the Srivastava-Singhal polynomials
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 \(L_n^{(\alpha)}(x)\).
scientific article; zbMATH DE number 2107275