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Well‐posed fractional calculus operator: Obtaining new transformations formulas involving Gauss hypergeometric functions with rational quadratic, cubic, and higher degree arguments

In this paper, we propose a systematic method for discovering new transformation formulas for the Gauss hypergeometric function with quadratic and rational (quadratic, cubic, and of higher degree) arguments. These new transformation formulas are obtained from known transformation formulas given in 1881 by Goursat (E. Goursat, Sur l'Équation différentielle linéaire qui admet pour intégrale la série hypergéométrique, Annales scientifique de l'É. N. S., 2e série tome 10 [1881], 3–142). This method relies on the use of the well‐posed fractional calculus operator g(z)Oβα introduced by Tremblay (R. Tremblay, Une contribution à la théorie de la dérivée fractionnaire, Doctoral thesis, Université Laval, Québec, Canada [1974]). We illustrate the effectiveness of the method by giving several presumably new transformation formulas for the Gauss hypergeometric function.