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We sum multivariate generating functions composed of products of Chebyshevpolynomials of the first and the second kind. That is, we find closed forms ofexpressions 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 withHermite polynomials replaced by Chebyshev polynomials of the first or thesecond kind. In all the considered cases, the obtained closed forms arerational functions with positive denominators. We show how to apply theobtained results to integrate some rational functions or sum some relatedseries of Chebyshev polynomials. We hope that the obtained formulae will beuseful in the so-called free probability. We expect also that the obtainedresults should inspire further research and generalizations. In particular,that, following methods presented in this paper, one would be able to obtainsimilar formulae for the so-called $q-$Hermite polynomials. Since the Chebyshevpolynomials of the second kind considered here are the $q$-Hermite polynomialsfor $q=0$. We have applied these methods in the one- and two-dimensional casesand were able to obtain nontrivial identities concerning $q-$Hermitepolynomials.

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