I am looking for the product of Gaussian hypergeometric functions of the form $_2F_1(a,b;c;\lambda z) _2F_1(d,e;f;\phi z)$. I would like, but do not expect, an answer of the form $_pF_q(\mathbf{a},\mathbf{b};(\lambda*\phi) z)$, where $\mathbf{a}$ is the numerator $p$vector and \mathbf{b} is the denominator $q$vector for the $(p,q)$hypergeometric function. I have not found relevant answers in the typical references, e.g., Slater.

For results of this sort, see Driver and Love, Products of hypergeometric functions and the zeros of 4F3 polynomials, and references therein (Numerical algorithms, 2001) 


I slightly misrepresented my problem. I wanted the product of ${}_2F_1(a,b;c;λz)\times {}_2F_1(d,e;c;ϕz)$. Note that the third argument $c$ is identical in the two functions and that the two final arguments are proportional $λz$ and $\phi z$. There appears to be no general solution to this product, although there are many restricted solutions. I had hoped my restrictions might have a solution, but apparently after reading Bailey's, Watson's, Preece's and Driver and Love's manuscripts, this is not to be. Srivastava provides a solution in terms of a double hypergeometric function, but I was hoping for something simpler. My posting contained a dreamedup form I would have liked. Ignore it. Sorry. Thanks for the Driver and Love reference. 

