Some incomplete hypergeometric functions and incomplete Riemann-Liouville fractional integral operators

Abstract

Very recently, the incomplete Pochhammer ratios were defined in terms of the incomplete beta function By(x, z). With the help of these incomplete Pochhammer ratios, we introduce new incomplete Gauss, confluent hypergeometric, and Appell's functions and investigate several properties of them such as integral representations, derivative formulas, transformation formulas, and recurrence relations. Furthermore, incomplete Riemann-Liouville fractional integral operators are introduced. This definition helps us to obtain linear and bilinear generating relations for the new incomplete Gauss hypergeometric functions.