# Article ID: MTJPAM-D-20-00057

## Title: Using the Well-Poised Fractional Calculus Operator ${}_{g(z)} O_{\beta}^{\alpha}$${}_{g(z)} O_{\beta}^{\alpha}$ to obtain transformations of the Gauss hypergeometric function with higher level arguments

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

Article ID: MTJPAM-D-20-00057; Volume 3 / Issue 3 / Year 2021 (Special Issue), Pages 260-283

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: 30 December 2020, Accepted: 6 March 2021, Published: 25 April 2021.

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

Full Text: PDF

Abstract

The main objective of this article is to add twelve new transformations formulas for the Gauss hypergeometric function having higher-order rational arguments than those recently obtained by Tremblay (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), p. 36 – 62, 2020) and Tremblay and Gaboury (R.Tremblay and S. Gaboury, Well-posed fractional calculus: obtaining new transformations formulas involving Gauss hypergeometric functions with rational quadratic, cubic and higher degree arguments, Math. Meth. Appl. Sc., (13) (2018), p. 4967-4985). These transformation formulas are obtained with a new systematic method applied to known formulas, most of which come from the Goursat thesis published in 1881 (E. Goursat, Sur l’Équation différentielle linéaire qui admet pour intégrale la série hypergéométrique, Annales scientifiques de l’É. N. S., 2e série tome 10 (1881), 3–142). The method used is based on the use of the fractional operator ${}_{g(z)} O_{\beta}^{\alpha}$ called ‘well-poised fractional calculus operator’ introduced a long time ago by Tremblay (R. Tremblay, Une contribution à la théorie de la dérivée fractionnaire, Doctoral thesis, Université Laval, Québec, Canada (1974)). After presenting the definition and a short list of these properties of the operator  ${}_{g(z)} O_{\beta}^{\alpha}$, we give an detailed example of of calculations to obtain this type of transformation.

Keywords: Fractional derivatives, Well-poised fractional calculus operator, Special functions, Gauss hypergeometric function, Transformation formulas

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