${}_3{F}_4$ hypergeometric functions as a sum of a product of ${}_2{F}_3$ functions

This paper shows that certain ${}_3{F}_4$ hypergeometric functions may be expanded in sums of pair products of ${}_2{F}_3$ 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. In addition to its intrinsic value, this result has a specific application in calculating the response of the atoms to laser stimulation in the Strong Field Approximation.