Is there any formula that gives the following specific product of two Hypergeometric functions $$_2 F_1(\alpha, \alpha+1 ; m ; z)\, _2 F_1(\alpha, \alpha+1 ; n ; z)\, , $$ for $m, n $ positive integers and $\alpha$ a positive half integer?

I am aware of the general formula in terms of a sum of $_4 F_3$'s, but I am looking for a formula that contains just a single $_p F_q$ which should exist according to "Higher Transcendental Functions, Vol. 1" by A. Erdelyi page 185.

P.S.

There is some physics behind this problem. It is a piece of a kernel in the energy basis arising in the study of a problem in Quantum Mechanics.