On Some Expansion Theorems Involving Confluent Hypergeometric $_{2}F_{2}(x)$ Polynomial

TL;DR

This paper extends transformation techniques for confluent hypergeometric $_{2}F_{2}(x)$ polynomials to derive new expansion theorems, including known and novel results, enhancing the understanding of these special functions.

Contribution

It introduces a new Kummer-type transformation for $_{2}F_{2}(a, d+2; b, d; x)$ and develops novel expansion theorems based on this transformation.

Findings

Derived new expansion theorems for $_{2}F_{2}(x)$

Reproduced known results as special cases

Presented several new particular case results

Abstract

Recently, Rathie and K{\i}l{\i}\c{c}man (2014) employed Kummer-type transformation for $_{2}F_{2}(a, d+1; b, d; x)$ to develop certain classes of expansions theorems for $_{2}F_{2}(x)$ hypergeometric polynomial. Our aim is to deduce Kummer-type transformation for $_{2}F_{2}(a, d+2; b, d; x)$ and utilize it to develop some new expansion theorems for the confluent hypergeometric $_{2}F_{2}(x)$ polynomial. We also obtain a well-known result given by Kim et al. (Integral Transforms Spec. Funct. 23(6); 435-444, 2012) and many other new results as particular cases of our theorems.