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.

$\begingroup$ I am having trouble parsing this. Are you trying to express the product of two $_2F_1$s as a $_pF_q?$ $\endgroup$– Igor RivinJun 12, 2012 at 18:55

$\begingroup$ Can you expand on what you really want? One can come up with restrictions on the parameters to make your wish come true, but is that what you are after? In the general case, all you can really say is that the answer will be a solution of a (huge) linear ODE of order 4 (and degree 11, with dense coefficients in a,b,c,d,e,f,$\lambda$,$\phi$). $\endgroup$– Jacques CaretteJun 13, 2012 at 0:28
2 Answers
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.