3F4 Hypergeometric Functions as a Sum of a Product of 1F2 Functions

This paper shows that certain F43 hypergeometric functions can be expanded in sums of pair products of F21 functions. In special cases, the F43 hypergeometric functions reduce to F32 functions. Further special cases allow one to reduce the F32 functions to F21 functions, and the sums to products of F10 (Bessel) and F21 functions. The class of hypergeometric functions with summation theorems are thereby expanded beyond those expressible as pair-products of F12 functions, F23 functions, and generalized Whittaker functions, into the realm of Fqp functions where p<q for both the summand and terms in the series.