On a class of transformations of sequences of complex numbers

TL;DR

This paper studies a transformation of complex number sequences, finds its inverse, connects it to the binomial transform, and derives new relations among hypergeometric orthogonal polynomials and series.

Contribution

It introduces a generalized transformation of sequences, finds its inverse, and establishes new relations among classical hypergeometric orthogonal polynomials.

Findings

Derived the inverse of the transformation $L_a$ and related $ ilde{L}_a$.

Connected the transformation to the binomial transform.

Established new formulas for sums of terminating hypergeometric series.

Abstract

In this paper we consider a transformation $L_a$ of sequences of complex numbers. We find the inverse transformation of $L_a$ as well as the inverse of a related transformation $\tilde{L}_a$. We explore a connection to the binomial transform and significantly generalize a previously known result. We also obtain new relations among many classical hypergeometric orthogonal polynomials as well as new formulas for sums involving terminating hypergeometric series.