Decomposition of some hypergeometric polynomials with respect to the cyclic group of order $n$

Abstract

Let $\left\{ P_{m}\right\} _{m\geq0}$ be a sequence of polynomials with complex coefficients and let n be an arbitrary positive integer. The components with respect to the cyclic group of order n of the polynomial $P_{m},m=0,1,...,$ are given by: \[ \left(P_{m}\right)_{\left[n,k\right]}\left(z\right)=\frac{1}{n}\overset{n-1}{\overset{\sum}{l=0}}\;\omega_{n}^{-kl}P_{m}\left(\omega_{n}^{l}z\right)\:,\quad k=0,1,...,n-1\;, \] where $\omega_{n}=exp\left(\frac{2i\pi}{n}\right)$. In this paper, we consider two class of hypergeometric polynomials, the Brafman polynomials and the Srivastava-Panda polynomials. For the components of these polynomials, we establish hypergeometric representations, differential equations and generating functions.