Recently, many researchers are interested in the investigation of an extended form of special functions like Gamma function, Beta function, Gauss hypergeometric function, Confluent hypergeometric function and Mittag-Leffler function etc. Here, in this paper, the main objective is to find the composition of Caputo MSM fractional differential of the extended form of Mittag-Leffler function in terms of extended Beta function. Further, in this sequel, some corollaries and consequences are shown that are the special case of our main findings.
Popular literature of Special Functions includes the generalization and extensions of functions like gamma function, beta function, Mittag-Leffler function, hypergeometric function and confluent hypergeometric function etc. This sequel deals with the extension of Mittag-Leffler functions and its properties. We aim to find the composition of fractional integration formula known as Ҏ − transform with the extended Mittag-Leffler function. Some special cases and corollaries are pointed out which follow from our main results.
This research proves the existence of the solution for the Fredholm integral equation of the first kind. Initially, k−Riemann integral
equation is considered involving the k−hypergeometric function as kernel. k−fractional integration defined by Mubeen and Habibullah [16] is
used to investigate the solution of the integral equation
Z x
0
(x − t)
c
k −1
Γk(c)
q+1Fq,k Ã
(ai
, k),(b, k)
(ci
, k)
; 1 −
x
t
!
f(t)dt = g(x)
where λ, ai
, b, ci > 0, i = 1, . . . , q and f ∈ C◦.
To prove the existence of solution, necessary and sufficient conditions are
defined.
Keywords: k−Pochhammer symbol, k−hypergeometric function, k−Fractional
Integration, k−Riemann integral equation, Fredholm integral equation
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