A NEW CHARACTERIZATION OF 𝑞-CHEBYSHEV POLYNOMIALS OF THE SECOND KIND

Abstract

In this work, we introduce the notion of $\cal{U}_{(q, \mu)}$-classical orthogonal polynomials, where $\cal{U}_{(q, \mu)}$ is the degree raising shift operator defined by $\cal{U}_{(q, \mu)}$ $:= x(xH_q + q^{-1}I_{\cal{P}}) + \mu H_q$, where $\mu$ is a nonzero free parameter, $I_{\cal{P}}$ represents the identity operator on the space of polynomials $\cal{P}$, and $H_{q}$ is the q-derivative one. We show that the scaled q-Chebychev polynomials of the second kind $\hat{U}_{n}(x, q), n\geq 0$, are the only $\cal{U}_{(q, \mu)}$-classical orthogonal polynomials.

Keywords

• orthogonal 𝑞-polynomials
• 𝑞-derivative operator
• 𝑞- chebyshev polynomials
• raising operator