Article ID: MTJPAM-D-22-00024

Title: Some examples of application of the operator zOβα to special functions, in particular to the Christoffel-Darboux identity for orthogonal polynomials

Montes Taurus J. Pure Appl. Math. / ISSN: 2687-4814

Article ID: MTJPAM-D-22-00024; Volume 5 / Issue 3 / Year 2023, Pages 67-94

Document Type: Research Paper

Author(s): Richard Tremblay a

aDépartement d’Informatique et Mathématique, Université du Québec à Chicoutimi, Chicoutimi, Qué., Canada G7H 2B1

Received: 18 July 2021, Accepted: 11 August 2022, Published: 26 November 2022.

Corresponding Author: Richard Tremblay (Email address: rtrembla@uqac.ca)

Full Text: PDF

Abstract

In the field of special functions, the theory relating to sequences of orthogonal polynomials functions and multiple orthogonal polynomials is fundamental. There are several formulas and many useful applications in mathematical physics, numerical analysis, statistics and probability and in many other disciplines. For example, the well-known identity of Christoffel-Darboux has generated a large number of research articles. Recently, Tygert (Analogues for Bessel functions of Christoffel-Darboux identity, Research Report Yaleu/Dcs/Rr-1351, 1–8, 2006) obtained two similar new identities for Bessel functions according to the well-known Christoffel-Darboux formula. This identity has even been generalized for orthogonal matrix polynomials (E. Daems and A. B. J. Kuijlaars, A Christoffel–Darboux formula for multiple orthogonal polynomials, J. Approx. Theory 130, 188–200, 2004). In this article, we obtain several summation formulas involving the classical orthogonal polynomials (Sections 7 and 8) using the well-balanced fractional operator defined in terms of fractional derivative, $_{z}O^{\alpha}_{\beta}f(z)=\frac{\Gamma(\beta)}{\Gamma(\alpha )} z^{1-\beta}D_{z}^{\alpha-\beta}z^{\alpha-1}f(z)$ (Section 2). This operator has several operational properties (Section 3) and it has already been used in several papers involving special functions, for example obtaining several new higher-order transformations of the Gaussian hypergeometric function (R. Tremblay, New quadratic transformations of hypergeometric functions and associated summation formulas obtained with the well-poised fractional calculus operator, Montes Taurus J. Pure Appl. Math. 2 (1), 36–62, 2020 and R. Tremblay, S. Gaboury, Well-posed fractional calculus: obtaining new transformations formulas involving Gauss hypergeometric functions with rational quadratic, cubic and higher degree arguments, Math. Methods Appl. Sci. 41 (13), 4967–4985, 2018). Firstly, we demonstrate unequivocally using examples the efficiency of the fractional operator zOβαf(z) to generate new relations involving the special functions of one or more variables. As for the fractional derivative, these functions can be represented in several forms using this operator (Section 4 and Tables A.1, A.2, A.3, A.4. Second, this operator also offers the possibility of discovering new avenues of research. We prove this in Section 5 and Section 6 where we obtain new formulas involving the generalized hypergeometric function and an extension of the generalized Bernoulli polynomials. Thirdly, we apply the operator zOβαf(z) to the classical Christoffel-Darboux identity and we obtain two general summation formulas for orthogonal polynomials (Theorem 7.1 and Corollary 7.2). Section 7 explicitly explores these formulas for the main orthogonal polynomials. Finally, a generalization of the formulas for any family of functions satisfying a three-term symmetric recurrence formula is given in Section 8.

Keywords: Fractional derivatives, Well-Poised fractional calculus operator, special functions, generalized hypergeometric functions, orthogonal polynomials, Christoffel-Darboux identity, Bessel functions, generalized Bernoulli polynomials, summation formulas

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