Abstract
Recently, various forms of extended beta function have been proposed and presented by many researchers. The principal goal of this paper is to present another expansion of beta function using Appell series and Lauricella function and examine various properties like integral representation and summation formula. Statistical distribution for the above extension of beta function has been defined, and the mean, variance, moment generating function and cumulative distribution function have been obtained. Using the newly defined extension of beta function, we build up the extension of hypergeometric and confluent hypergeometric functions and discuss their integral representations and differentiation formulas. Further, we define a new extension of Riemann–Liouville fractional operator using Appell series and Lauricella function and derive its various properties using the new extension of beta function.
Highlights
140] defined an extension of beta function as follows: Bγp1,q,γ2 (η1, η2) =
The classical beta function is given by [1, Eq (16), p. 18]B( 1, 2) = t 1–1(1 – t) 2–1 dt =( 1) ( 2), ( 1 + 2) (1)where ( 1), ( 2) > 0, is the real part of the function
6 Extension of Riemann–Liouville fractional operators we extend the Riemann–Liouville fractional operators using the Appell series and derive its properties using the new extension of beta function
Summary
140] defined an extension of beta function as follows: Bγp1,q,γ2 (η1, η2) = Definition 2.1 The extensions of beta function using Appell series (10)–(13), respectively, are defined as follows: (a) = F3 a1, a1, a2, a2; a3; tr , 0 = F4 a1, a2; a3, a3; tr , 0 p = 2F1 a1, a2; a3; tr , for q = 0 in Definition 2.1, we obtain the following result: The extension of beta function using hypergeometric series is defined as follows: Bp( 1, 2) = Definition 2.2 The extensions of beta function using Lauricella series (14)–(17) respectively, are defined as follows: (a) BFp1A(n,.)..,pn ( 1, 2)
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