Communications of the
Korean Mathematical Society CKMS

In this present paper, we consider four integrals and differentials containing the Gauss' hypergeometric ${}_2{F}_1(x)$ function in the kernels, which extend the classical Riemann-Liouville (R-L) and Erd\'elyi-Kober (E-K) fractional integral and differential operators. Formulas (images) for compositions of such generalized fractional integrals and differential constructions with the $n$-times product of the generalized modified Bessel function of the second type are established. The results are obtained in terms of the generalized Lauricella function or Srivastava-Daoust hypergeometric function. Equivalent assertions for the Riemann-Liouville (R-L) and Erd\'elyi-Kober (E-K) fractional integral and differential are also deduced.

Keywords: Saigo fractional calculus operators, generalized Lauricella function, Gauss' hypergeometric ${}_2{F}_1(x)$ function, generalized modified Bessel function of the second type

MSC numbers: 26A33, 33C10, 33C20, 33C50, 33C60, 26A09

Supported by: This work is supported by the Competitive Research Scheme (CRS) project funded by the TEQIP-III (ATU) Rajasthan Technical University Kota under grant number TEQIP-III/RTU(ATU)CRS/2019-20/50.