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may 2021 Jacobi polynomials and some connection formulas in terms of the action of linear differential operators
Baghdadi Aloui, Jihad Souissi
Bull. Belg. Math. Soc. Simon Stevin 28(1): 39-51 (may 2021). DOI: 10.36045/j.bbms.200606

Abstract

In this paper, we introduce the concept of the $\mathbb{J}_{\alpha, \beta}$-classical orthogonal polynomials, where $\mathbb{J}_{\alpha, \beta}$ is the raising operator $\mathbb{J}_{\alpha, \beta}:=(x^2-1)\frac{d}{dx}+ \big((\alpha+\beta)x-\alpha+\beta\big)\mathbb{I}$, with $\alpha$ and $\beta$ nonzero complex numbers and $\mathbb{I}$ representing the identity operator. Then, we show that the Jacobi polynomials $P^{(\alpha, \beta)}_n(x),\ n\geq0$, where $\alpha, \beta\in\mathbb{C}\backslash\{0,-1,-2,\ldots\}$, $(\alpha+\beta\neq-m, \ m\geq0)$, are the only $\mathbb{J}_{\alpha, \beta}$-classical orthogonal polynomials. As an application, we give some new connection formulas satisfied by the polynomials solution of our problem.

Citation

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Baghdadi Aloui. Jihad Souissi. "Jacobi polynomials and some connection formulas in terms of the action of linear differential operators." Bull. Belg. Math. Soc. Simon Stevin 28 (1) 39 - 51, may 2021. https://doi.org/10.36045/j.bbms.200606

Information

Published: may 2021
First available in Project Euclid: 2 June 2021

Digital Object Identifier: 10.36045/j.bbms.200606

Subjects:
Primary: 33C45
Secondary: 42C05

Rights: Copyright © 2021 The Belgian Mathematical Society

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Vol.28 • No. 1 • may 2021
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