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

作者：Tremblay R; Gaboury S^*

来源：Mathematical Methods in the Applied Sciences, 2018, 41(13): 4967-4985.

DOI：10.1002/mma.4945

• fractional calculus
• Gauss hypergeometric function
• special functions of mathematical physics
• transformation formula

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'Equation differentielle lineaire qui admet pour integrale la serie hypergeometrique, Annales scientifique de l'E. N. S., 2e serie tome 10 [1881], 3-142). This method relies on the use of the well-posed fractional calculus operator g(z)O introduced by Tremblay (R. Tremblay, Une contribution a la theorie de la derivee fractionnaire, Doctoral thesis, Universite Laval, Quebec, Canada [1974]). We illustrate the effectiveness of the method by giving several presumably new transformation formulas for the Gauss hypergeometric function.

• 出版日期2018-9-15