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