Multivariate generating functions built of Chebyshev polynomials and some of its applications and generalizations

Abstract: We sum multivariate generating functions composed of products of Chebyshev polynomials of the first and the second kind. That is, we find closed forms of expressions of the type $\sum_{j\geq0}\rho^{j}\prod_{m=1}^{k}T_{j+t_{m}}% (x_{m})\prod_{m=k+1}^{n+k}U_{j+t_{m}}(x_{m}),$ for different integers $t_{m},$ $m=1,...,n+k.$ We also find a Kibble-Slepian formula of $n$ variables with Hermite polynomials replaced by Chebyshev polynomials of the first or the second kind. In all the considered cases, the obtained closed forms are rational functions with positive denominators. We show how to apply the obtained results to integrate some rational functions or sum some related series of Chebyshev polynomials. We hope that the obtained formulae will be useful in the so-called free probability. We expect also that the obtained results should inspire further research and generalizations. In particular, that, following methods presented in this paper, one would be able to obtain similar formulae for the so-called $q-$Hermite polynomials. Since the Chebyshev polynomials of the second kind considered here are the $q$-Hermite polynomials for $q=0$. We have applied these methods in the one- and two-dimensional cases and were able to obtain nontrivial identities concerning $q-$Hermite polynomials.