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



Sum over Hypergeometric function 2F1 (generating function)
The wiki page on [Legendre polynomials](http://en.wikipedia.org/wiki/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
$$\sum\_{n\geq 0}\sum\_{k=0}^n \binom{n}{k}^2 x^ny^k=\sum\_{n\geq 0} x^n(y-1)^n P\_n\left(\frac{y+1}{y-1}\right)$$
$$=\frac{1}{\sqrt{1-2x(y+1)+x^2(y-1)^2}}.$$


