Abstract
Very recently, the incomplete Pochhammer ratios were defined in terms of the incomplete beta function B y ( 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.
Highlights
Introduction and PreliminariesIn recent years, some extensions of the well-known special functions have been considered by several authors
We investigate the nth derivatives of the incomplete beta function by means of incomplete Pochhammer ratios
In [27], various applications of fractional calculus were exhibited in areas ranging from engineering to life sciences
Summary
Some extensions of the well-known special functions have been considered by several authors (see, for example, [1,2,3,4,5,6,7,8,9]). Fractional derivative and integral operators are another important topic of research in recent years. The use of fractional derivative operators in obtaining generating relations for some special functions can be found in [6,9,28,29,30]. In a recent paper [12], which covered work done after the work we introduced incomplete Liouville-Caputo fractional derivative operators and focused on their use in special function theory. We introduce incomplete Riemann-Liouville fractional integral operators, and we obtain some generating relations for these new incomplete hypergeometric functions with the aid of these new defined operators. In the last section, we obtain linear and bilinear generating relations for the incomplete hypergeometric functions
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