Multivariate generating functions involving Chebyshev polynomials

Paweł J. Szabłowski

Published 2017-06-01Version 1

We sum multivariate generating functions composed of mixtures of Chebyshev polynomials of the first and second kind. That is we find closed form of expressions of the type $\sum_{j\geq 0}\rho ^{j}\prod_{m=1}^{k}T_{j+t_{m}}(x_{m})\prod_{m=k+1}^{n+k}U_{j+t_{m}}(x_{m}),$ for different integers $t_{m},$ $m=1,...,n+k..$ We also sum Kibble-Slepian formula of $n$ variables with Hermite polynomials replaced by Chebyshev polynomials of the first and the second kind. In all those cases that were considered, the obtained closed forms are of the form of rational functions with positive denominators. We perform all this calculations basically in order to simplify certain multivariate integrals or obtain multivariate distribution having compact support as well as for pure curiosity.