Sum over Hypergeometric function 2F1 (generating function)

Question

Dear mathematicians,

in my current research project I came accross this very bothersome sum over a rather simple hypergeometric function, or formulated differently: a sum over squared binomial coefficients:

$\sum_{b=0}^\infty Y^b\ _2F_1(-b,-b;1;X)=\sum_{b=0}^\infty \sum_{k=0}^\infty \binom{b}{k}^2 Y^bX^k$

I was wondering, if there exists a generating function for these particular sums? In the case where the binomial coefficient is not squared this is simply $\frac{1}{1-Y-XY}$, but unfortuntely I have not found anything on the above problem.

Any help concerning this question would be deeply appreciated. Have a good weekend, jan

Answer 1

The wiki page on Legendre polynomials has all the identities that I will use below.

Start with $$P_n(t)=\sum_{k=0}^n (-1)^k \binom{n}{k}^2 \left(\frac{1+t}{2}\right)^{n-k}\left(\frac{1-t}{2}\right)^k$$ which can be rewritten as $$(t-1)^nP_n\left(\frac{t+1}{t-1}\right)=\sum_{k=0}^n \binom{n}{k}^2 t^k,$$ so that $$\sum_{n\geq 0}\sum_{k=0}^n \binom{n}{k}^2 x^ny^k=\sum_{n\geq 0} x^n(y-1)^n P_n\left(\frac{y+1}{y-1}\right)$$ $$=\frac{1}{\sqrt{1-2x(y+1)+x^2(y-1)^2}}.$$