$\,_{3}F_{4}$ hypergeometric functions as a sum of a product of $\,_{2}F_{3}$ functions

TL;DR

This paper demonstrates how certain $_{3}F_{4}$ hypergeometric functions can be expressed as sums of products of $_{1}F_{2}$ functions, expanding the known classes of hypergeometric functions with summation theorems.

Contribution

It introduces new expansions of $_{3}F_{4}$ functions into sums of $_{1}F_{2}$ products, broadening the scope of hypergeometric functions with summation formulas.

Findings

$_{3}F_{4}$ functions can be expanded as sums of $_{1}F_{2}$ products.

Special cases reduce $_{3}F_{4}$ to $_{2}F_{3}$ and further to $_{1}F_{2}$ functions.

The work extends summation theorems to a wider class of $_{p}F_{q}$ functions.

Abstract

This paper shows that certain $\,_{3}F_{4}$ hypergeometric functions can be expanded in sums of pair products of $\,_{1}F_{2}$ functions. In special cases, the $\,_{3}F_{4}$ hypergeometric functions reduce to $\,_{2}F_{3}$ functions. Further special cases allow one to reduce the $\,_{2}F_{3}$ functions to $\,_{1}F_{2}$ functions, and the sums to products of $\,_{0}F_{1}$ (Bessel) and $\,_{1}F_{2}$ functions. This expands the class of hypergeometric functions having summation theorems beyond those expressible as pair-products of generalized Whittaker functions, $\,_{2}F_{1}$ functions, and $\,_{3}F_{2}$ functions into the realm of $\,_{p}F_{q}$ functions where $p<q$ for both the summand and terms in the series.