Where can I find generating functions for orthogonal polynomials in two variables? Lebedev's book (Special Functions and their Applications, Dover, 1972) gives a closed form for $$ \sum_{n=0}^\infty \frac{n!}{\Gamma(n+\alpha+1)} L_n^\alpha(x) L_n^\alpha(y) t^n $$ as an explicit function of $x,y,t$ (it involves a Bessel function). However, I need to sum over the upper index (when it is an integer), to find $$ \sum_{k=0}^\infty \frac{n!}{(n+k)!} L_n^k(x) L_n^k(y) t^k = {??} $$ in a closed form $F(x,y,t)$, for a fixed positive integer $n$. Are generating functions of this kind to be found in the literature? (Not on DLMF, as far as I can see.) A closed form for the previous sum would be great, a pointer to a suitable article would be better.
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I think that http://arxiv.org/pdf/math-ph/0409066v1 (Multivariate Orthogonal Polynomials (symbolically) page 15, has the representation you are looking for [whether it will help you compute your sum, I am not sure, but maybe their Maple package will do the thinking for you?
answered Dec 9, 2011 at 13:47
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$\begingroup$ Not really, but its bibliography pointed to an article of Baker and Forrester (Commun. Math. Phys. 188, 1997) that has a nice general theory of multivariate generating functions. Sadly, nothing there about summation on the upper index. $\endgroup$– jvarillyCommented Dec 9, 2011 at 14:47