Is there a relationship or recurrence relation I can use to solve for
$$\, _2F_1(b,r+k;a+b+k;p)$$
as a function of $k$, with known value of when $k=0$
$$ \, _2F_1(b,r;a+b;p) = f_0$$
(a,b,r,p) are positive real numbers and k is an integer.
2026-03-27 18:07:11.1774634831
Hypergeometric function relation knowing initial value?
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Yes, the formula 15.5.7 here (after exchanging $a$ and $b$). That is, \begin{align} &_2F_1(a,b+k,c+k,z)=\\=&(-1)^k\frac{(c)_k(1-z)^{1-b-k}}{(c-a)_k(b)_k}\left((1-z)\frac{d}{dz}(1-z)\right)^k (1-z)^{b-1}{}_2F_1(a,b,c,z), \end{align} where $(s)_k$ denotes the Pochhammer symbol.