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