Extended Fractional Hypergeometric Function and Applications

Hypergeometric functions (Beta and Gamma functions) are a very important family member of the special functions and they play a vital role in the whole theory of special functions. Hypergeometric functions together with their extension have many applications in research fields such as engineering, chemical, statistics, fractional calculus, and physical problems. In this talk, we have focused on the extended Euler’s beta function, which is developed by using the 2-parameter Mittag-Leffler function as the kernel. We discuss various basic properties and formulas of the extended Euler’s beta function such as integral representations, transformation formulas, and summation formulas. We also introduce the logarithmic convexity and some important inequalities for this extended Euler’s beta function. Then by using this extended Euler’s beta function as a kernel, we have generalized hypergeometric functions and study various properties of these extended hypergeometric functions. From the application point of view, we have also derived some relations between this extended Euler’s beta function and extended fractional derivative operators such as Caputo fractional derivative operator and Riemann-Liouville fractional operators.
Keywords: Beta function, Gamma function, fractional calculus