Abstract
In the present paper author derive a number of integrals concerning various special functions which are applications of the one of Osler result. Osler provided extensions to the familiar Leibniz rule for the nth derivative of product of two functions.
Highlights
In recent years there has appeared a great deal of literature discussing the application of the aforementioned fractional calculus operators in a number of areas of mathematical analysis and stands on fairly firm footing through the research contribution of various authors
The integral analog of the generalized Leibnitz rule in the theory of fractional calculus was given by Osler [19] in the following form:
In view of the large number of parameters involved in the results established above, these integrals are capable of yielding a number of known results and new interesting results including integrals due to Arora and Koul [13, p.932, Eqs. (2.1) and (2.2)]
Summary
In recent years there has appeared a great deal of literature discussing the application of the aforementioned fractional calculus operators in a number of areas of mathematical analysis (such as ordinary and partial differential equations, integral equations, summation of series, et cetera) and stands on fairly firm footing through the research contribution of various authors (cf., e.g., [1,2,3,4,5,6, 7,8,9,10,11,12,14,16,17] ). The integral analog of the generalized Leibnitz rule in the theory of fractional calculus was given by Osler [19] in the following form: Dzα−η [u(z)] Dzη [v(z)] (α, η ∈ C) The generalized hypergeometric function of one variable viz., pFq[.; .; z] defined as (see, for details, Srivastava and Karisson 1985, [20, p.19]): pFq[z] = pFq (ap); (bq ); z n=0 pj=1(aj )n q j=1
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