<s>[INST] How to speedup evaluation of hypergeometric ${}_3 F_2(1)$? I need to make a table of ${}_3 F_2\left(\frac{a}2+\frac14, \frac{a}2+\frac34, \frac{a+b}2;\; a+1,\frac{a+b}2+1;\;1\right)$ for integer $a, b,$ $0\le a\le N_1$, $0\le b\le N_2$, with precision of 50 decimal places in mantissa.
Currently I'm trying to use Wolfram Mathematica for this task, but it appears to evaluate a single value for over a minute when $a$ and $b$ are $\sim50$. And as $N_1\approx50$ and $N_2\approx 300$, this appears to be too long a computation.
So I'm looking for ways to speed up this calculation. Namely, is there a way to reduce this function to something simpler for these particular arguments? Are there some relations which would allow to easily compute it for given $a_i, b_i$, having computed for several other values (like the relation $\Gamma(x+1)=x\Gamma(x)$, which lets one easily find $\Gamma(x+1)$ knowing $\Gamma(x)$ without computing $\Gamma$ once more)?
 [/INST] Hint:
$_3F_2\left(\dfrac{a}{2}+\dfrac{1}{4},\dfrac{a}{2}+\dfrac{3}{4},\dfrac{a+b}{2};a+1,\dfrac{a+b}{2}+1;1\right)$
$=\sum\limits_{n=0}^\infty\dfrac{\left(\dfrac{a}{2}+\dfrac{1}{4}\right)^{(n)}\left(\dfrac{a}{2}+\dfrac{3}{4}\right)^{(n)}\left(\dfrac{a+b}{2}\right)^{(n)}}{(a+1)^{(n)}\left(\dfrac{a+b}{2}+1\right)^{(n)}n!}$
$=\sum\limits_{n=0}^\infty\dfrac{\left(\dfrac{a}{2}+\dfrac{1}{4}\right)^{(n)}\left(\dfrac{a}{2}+\dfrac{3}{4}\right)^{(n)}\dfrac{a+b}{2}}{(a+1)^{(n)}\left(n+\dfrac{a+b}{2}\right)n!}$
$=\dfrac{a+b}{2}\int_0^1\sum\limits_{n=0}^\infty\dfrac{\left(\dfrac{a}{2}+\dfrac{1}{4}\right)^{(n)}\left(\dfrac{a}{2}+\dfrac{3}{4}\right)^{(n)}x^{n+\frac{a+b}{2}-1}}{(a+1)^{(n)}n!}~dx$
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