A new Riemann–Liouville type fractional derivative operator and its application in generating functions

• Published in: Advances in difference equations 2018-05, Vol.2018 (1), p.1-16, Article 167
• Main Authors: Shadab, M., Khan, M. Faisal, Lopez-Bonilla, J. Luis
• Format: Article
• Language: English
• Subjects: Analysis; Difference and Functional Equations; Extended Appell functions; Extended beta function; Extended Lauricella function; Functional Analysis; Generating function; Hypergeometric functions; Mathematical analysis; Mathematics; Mathematics and Statistics; Mellin Transform; Mellin transforms; Operators (mathematics); Ordinary Differential Equations; Partial Differential Equations; Riemann–Liouville fractional derivative operator
• ISSN: 1687-1847; 1687-1839; 1687-1847
• DOI: 10.1186/s13662-018-1616-9

Here, the concept of a new and interesting Riemann–Liouville type fractional derivative operator is exploited. Treatment of a fractional derivative operator has been made associated with the extended Appell hypergeometric functions of two variables and Lauricella hypergeometric function of three variables. With a view on analytic properties and application of new Riemann–Liouville type fractional derivative operator, we have obtained new fractional derivative formulas for some familiar functions and for Mellin transformation formulas. For the sake of justification of our new operator, we have established some presumably new generating functions for an extended hypergeometric function using the new definition of fractional derivative operator.