A panorama of generating functions for products of classical integer sequences

Abstract

In this note, we present the generating function
\[
\sum_{n=0}^\infty U_{n+\ell}(x)\, U_n(y)\, t^n = \frac{(1 - t^2) \, U_\ell(x) + 2  t (t x - y) \, U_{\ell-1}(x)}{1 - 4xy\, t + 2(2x^2 + 2y^2 - 1)t^2 - 4xy\, t^3 + t^4}\,,
\]
where \(U_n\) denotes the Chebyshev polynomial of the second kind of order $n$ and \(\ell\) is an integer. Although this generating function can be deduced from classical results, the matrix-based derivation presented here provides a unified and alternative perspective, drawing on lesser-known references. We give a concise historical overview from algebraic, trigonometric, combinatorial, and matrix perspectives for this type of products involving Chebyshev polynomials. Particular cases and extensions are discussed. Applications to recent identities involving products of various families of well-known number sequences are also presented.