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On the Orthogonality of the q-Derivatives of the Discrete q-Hermite I Polynomials

On the Orthogonality of the q-Derivatives of the Discrete q-Hermite I Polynomials
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Author(s): Sakina Alwhishi (El Mergib University, Libya), Rezan Sevinik Adıgüzel (Atılım University, Turkey) and Mehmet Turan (Atilim University, Turkey)
Copyright: 2020
Pages: 28
Source title: Emerging Applications of Differential Equations and Game Theory
Source Author(s)/Editor(s): Sırma Zeynep Alparslan Gök (Süleyman Demirel University, Turkey) and Duygu Aruğaslan Çinçin (Süleyman Demirel University, Turkey)
DOI: 10.4018/978-1-7998-0134-4.ch007

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Abstract

Discrete q-Hermite I polynomials are a member of the q-polynomials of the Hahn class. They are the polynomial solutions of a second order difference equation of hypergeometric type. These polynomials are one of the q-analogous of the Hermite polynomials. It is well known that the q-Hermite I polynomials approach the Hermite polynomials as q tends to 1. In this chapter, the orthogonality property of the discrete q-Hermite I polynomials is considered. Moreover, the orthogonality relation for the k-th order q-derivatives of the discrete q-Hermite I polynomials is obtained. Finally, it is shown that, under a suitable transformation, these relations give the corresponding relations for the Hermite polynomials in the limiting case as q goes to 1.

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