Sums of finite products of Chebyshev polynomials of two different types

&lt;abstract&gt;&lt;p&gt;In this paper, we consider sums of finite products of the second and third type Chebyshev polynomials, those of the second and fourth type Chebyshev polynomials and those of the third and fourth type Chebyshev polynomials, and represent each of them as linear combinations of Chebyshev polynomials of all types. Here the coefficients involve some terminating hypergeometric functions ${}_{2}F_{1}$. This problem can be viewed as a generalization of the classical linearization problems and is done by explicit computations.&lt;/p&gt;&lt;/abstract&gt;</jats:p

A new approach to Bell and poly-Bell numbers and polynomials

&lt;abstract&gt;&lt;p&gt;Bell polynomials are widely applied in many problems arising from physics and engineering. The aim of this paper is to introduce new types of special polynomials and numbers, namely Bell polynomials and numbers of the second kind and poly-Bell polynomials and numbers of the second kind, and to derive their explicit expressions, recurrence relations and some identities involving those polynomials and numbers. We also consider degenerate versions of those polynomials and numbers, namely degenerate Bell polynomials and numbers of the second kind and degenerate poly-Bell polynomials and numbers of the second kind, and deduce their similar results.&lt;/p&gt;&lt;/abstract&gt;</jats:p