Two-variable generating functions for Laguerre polynomials - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T21:59:41Z http://mathoverflow.net/feeds/question/83054 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/83054/two-variable-generating-functions-for-laguerre-polynomials Two-variable generating functions for Laguerre polynomials jvarilly 2011-12-09T13:10:31Z 2013-02-05T09:22:00Z <p>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.</p> http://mathoverflow.net/questions/83054/two-variable-generating-functions-for-laguerre-polynomials/83056#83056 Answer by Igor Rivin for Two-variable generating functions for Laguerre polynomials Igor Rivin 2011-12-09T13:47:31Z 2011-12-09T13:47:31Z <p>I think that <a href="http://arxiv.org/pdf/math-ph/0409066v1" rel="nofollow">http://arxiv.org/pdf/math-ph/0409066v1</a> (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?</p>