# Article ID: MTJPAM-D-20-00005

## Title: New Quadratic Transformations of Hypergeometric Functions and Associated Summation Formulas Obtained with the Well-Poised Fractional Calculus Operator

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

Article ID: MTJPAM-D-20-00005; Volume 2 / Issue 1 / Year 2020, Pages 36-62

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: 21 February 2020, Accepted: 24 April 2020, Available online: 9 May 2020.

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

Full Text: PDF

Abstract

Recently, Tremblay and Gaboury used the operator g(z)Oβα and obtained new formulas for the transformation of hypergeometric functions with higher order rational arguments (R.Tremblay and S. Gaboury, Well-poised 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). This operator was introduced for the first time in 1974 by Tremblay (R. Tremblay, Une contribution à la théorie de la dérivée fractionnaire [ Ph.D. thesis], Laval University, Quebec City, Canada). The main purpose of this article is to illustrate, using numerous examples, the efficiency of this operator in obtaining new results involving special functions. In particular, we deduce twenty-four new quadratic transformations of hypergeometric functions, and obtain several new summation theorems.

Keywords: Fractional derivatives, Well-Poised Fractional Calculus Operator, Special functions, Gauss hypergeometric function, Transformation formulas

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