Fractional calculus operators of the product of generalized modified Bessel function of the second type
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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.
Abstract
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.
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