Abstract
The aim of this paper is to deal with two integral transforms involving the Appell function as their kernels. We prove some compositions formulas for generalized fractional integrals with k-Bessel function. The results are expressed in terms of generalized Wright type hypergeometric function and generalized hypergeometric series. Also, the authors presented some related assertion for Saigo, Riemann-Liouville type, and Erdélyi-Kober type fractional integral transforms.
Highlights
Introduction and PreliminariesThe generalization of k-Bessel function is defined in Mondal [1] as Wξk,c (z) = ∞ ∑ n=0Γk (−c)n n!
We prove some compositions formulas for generalized fractional integrals with k-Bessel function
Special Cases of Theorem 2 (i) If we set υ = 0, μ = −γ, η = υ and replace υ by υ + μ in (18), we get the following corollary relating to left-hand sided Saigo fractional integral operator ([19, 20])
Summary
Where ξ > −1, k > 0, and c ∈ R and Γk(z) is the k-gamma function defined in Daaz and Pariguan [2] as. Special Cases of Theorem 2 (i) If we set υ = 0, μ = −γ, η = υ and replace υ by υ + μ in (18), we get the following corollary relating to left-hand sided Saigo fractional integral operator ([19, 20]). Special Cases of Theorem 6 (iv) If we substitute υ = 0, μ = −γ, η = υ and replace υ by υ + μ in (24), we get the subsequent corollary relating to right-hand sided Saigo fractional integral operator [19]. (v) If we set μ = −υ in (27), we get the following corollary relating to right-sided Weyl fractional type integral operator. (vi) If we set μ = 0 in (27), we get the subsequent corollary relating to right-hand side of Erdelyi-Kober fractional type integral operator. Let υ, γλ, ξ ∈ C be such that R(ξ) > −1, R(υ) > 0, R(λ + ξ) < 1 + min[0, R(γ)]; the following formula holds:
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