Multivariate generating functions built from Chebyshev polynomials and some of their applications and generalizations
Volume 169 / 2022
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 $\sum _{j\geq 0}\rho ^{j}\prod _{m=1}^{k}T_{j+t_{m}}(x_{m}) \times \prod _{m=k+1}^{n+k}U_{j+t_{m}}(x_{m}),$ for different integers $t_{m}$, $m=1,\ldots ,n+k$. We also find a Kibble–Slepian formula in $n$ variables with Hermite polynomials replaced by Chebyshev polynomials of the first or second kind. In all the cases considered, the closed forms obtained are rational functions with positive denominators. We show how to apply those results to integrate some rational functions or sum some related series of Chebyshev polynomials. We hope that the formulae obtained will be useful in free probability. We also expect that similar formulae can be obtained for $q$-Hermite polynomials. Since Chebyshev polynomials of the second kind are $q$-Hermite polynomials for $q=0$, we have applied our methods in the one- and two-dimensional cases and obtained nontrivial identities concerning $q$-Hermite polynomials.
