$\,_{3}F_{4}$ 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.