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ЖУРНАЛЫ // Проблемы анализа — Issues of Analysis // Архив

Пробл. анал. Issues Anal., 2024, том 13(31), выпуск 2, страницы 49–62 (Mi pa398)

A new characterization of $q$-Chebyshev polynomials of the second kind

S. Jbeli

Université de Tunis El Manar, Campus Universitaire El Manar, Tunis, 2092, Tunisie. LR13ES06

Аннотация: In this work, we introduce the notion of $\mathcal{U}_{(q,\mu)}$-classical orthogonal polynomials, where $\mathcal{U}_{(q,\mu)}$ is the degree raising shift operator defined by $\mathcal{U}_{(q,\mu)}:=x(xH_q+q^{-1}I_{\mathcal{P}})+\mu H_q,$ where $\mu$ is a nonzero free parameter, $I_{\mathcal{P}}$ represents the identity operator on the space of polynomials $\mathcal{P}$, and $H_q$ is the $q$-derivative one. We show that the scaled $q$-Chebychev polynomials of the second kind ${\hat{U}}_{n}(x, q), n\geq0$, are the only $\mathcal{U}_{(q,\mu)}$-classical orthogonal polynomials.

Ключевые слова: orthogonal $q$-polynomials, $q$-derivative operator, $q$-Chebyshev polynomials, raising operator.

УДК: 517.58

MSC: Primary 33C45; Secondary 42C05

Поступила в редакцию: 11.03.2024
Исправленный вариант: 26.05.2024
Принята в печать: 28.05.2024

DOI: 10.15393/j3.art.2024.15830



© МИАН, 2024