Abstract
The main aim of this paper is to give the definitions of Caputo fractional derivative operators and show their use in the special function theory. For this purpose, we introduce new types of incomplete hypergeometric functions and obtain their integral representations. Furthermore, we define incomplete Caputo fractional derivative operators and show that the images of some elementary functions under the action of incomplete Caputo fractional operators give a new type of incomplete hypergeometric functions. This definition helps us to obtain linear and bilinear generating relations for the new type incomplete Gauss hypergeometric functions.
Highlights
1 Introduction In recent years, some extensions of the well-known special functions have been considered by several authors
Proof Replacing the incomplete beta function By(a – m + n + k + r, e – a + m) in definition (30) by its integral representation given by (11), we find that
If we multiply both sides with zλ–1 and apply the incomplete Caputo fractional derivative operator Czλ–α, we get
Summary
Some extensions of the well-known special functions have been considered by several authors (see, for example, [4, 8, 9, 11, 15, 16, 18,19,20, 31]). Çetinkaya [7] introduced the incomplete second Appell hypergeometric functions by means of the incomplete Pochhammer symbols and obtained some integral representations and transformation formulas for these functions. After these works, incomplete hypergeometric functions have become one of the hot topics of recent years [4, 7, 11, 12, 14, 15, 21, 23, 25, 26, 29,30,31,32].
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