For $\Re a, \Re b, \Re c, \Re a', \Re b', \Re c'>0$, I would calculate the following product $$ {}_2 F_1(a, b; c; x^{-1}) \times \, {}_2 F_1(a', b'; c'; 1-\frac{x}{y}) $$ For all $y>x>1$. Is there a formula that allows me to calculate the above product
Thanks in advance.
This product of two Gaussian hypergeometric functions can be expressed by a sum over generalized hypergeometric functions ${}_{4}F_{3}$ according to formula 4.3.(14) on page 187 in "Higher Transcendental Functions, Vol. 1" by A. Erdelyi (Ed.). I reproduce it here for your convenience. (I've slightly changed the notation of the original.) $$ {}_{2}F_{1}(a,b;c;p z) \ {}_{2}F_{1}(a',b';c';q z)= \\ \sum_{n=0}^{\infty}\ \frac{(a)_{n} (b)_{n} (pz)^{n}}{n!\ (c)_{n}}\ {}_4F_{3}(a',b',1-c-n,-n;c',1-a-n,1-b-n;\frac{q}{p}), $$ where $(a)_{n}$ etc denotes the Pochhammer symbol. Note that for negative integer $a$ (and euivalently $b$ and per symmetry also for negative integer $a'$ and/or $b'$) the sum terminates. By setting $z=1$ and substituting $p$ and $q$ you get the solution that you are looking for.