3F4 Hypergeometric Functions as a Sum of a Product of 2F3 Functions

This paper shows that certain 3F4 hypergeometric functions can be expanded in sums of pair products of 2F3 functions, which reduce in special cases to 2F3 functions expanded in sums of pair products of 1F2 functions. This expands the class of hypergeometric functions having summation theorems beyond those expressible as pair-products of generalized Whittaker functions, 2F1 functions, and 3F2 functions into the realm of pFq 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.