The Pathway Fractional Integrals of Incomplete I -Functions
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The Pathway Fractional Integrals of Incomplete I-Functions D. Baleanu1,2,3 · N.K. Jangid4 · S. Joshi4 · S.D. Purohit5 Accepted: 18 September 2020 © Springer Nature India Private Limited 2020
Abstract In this present study, our aim is to investigate the pathway fractional integral formulae for the incomplete I -functions. Further, we also examine their special cases, which are in the form of recognize functions such as incomplete I -functions, incomplete H -functions and σ of incomplete H -functions. Findings connected with the fractional integral operator I0+ Riemann-Liouville type are also viewed as limiting cases of the main results. Keywords Pathway fractional integral operator · Incomplete I -functions · Incomplete I -functions · Incomplete H -functions · Incomplete H -functions · Riemann-Liouville fractional integral operator Mathematics Subject Classification 33B20 · 26A42 · 33D05
Introduction and Preliminaries It has been noted that there are many issues in the field of heat conduction and astrophysics where only the most specific classes of special functions are not essential to meet their answers to problems. In this case, the definition and analysis of incomplete Gamma functions and their generalizations have been included in the description. Related to incomplete gamma
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S.D. Purohit [email protected] D. Baleanu [email protected] N.K. Jangid [email protected] S. Joshi [email protected]
1
Department of Mathematics, Cankaya University, Etimesgut, Ankara, Turkey
2
Institute of space sciences, Magurele, Bucharest, Romania
3
Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan
4
Department of Mathematics & Statistics, Manipal University Jaipur, Jaipur, India
5
Department of HEAS (Mathematics), Rajasthan Technical University, Kota, India 0123456789().: V,-vol
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Int. J. Appl. Comput. Math
(2020) 6:151
functions, incomplete hypergeometric functions, incomplete H -functions, and numerous other generalized functions have been studied, see [9,17,23,24] and references therein. This prompted a number of researchers in the field of incomplete theory to investigate potential extensions of all relevant results, including numerous classical results. For example, one can refere to the recent papers [3–6,18]. In our current study we use incomplete I -functions, the generalized form of I -function described by Rathie [20], which are the expansion formulae of well recognized Fox’s H function [8] and many related functions. Recently, new classes of incomplete I -functions I m,n (z) and γ I m,n (z) have investigated by Jangid et al. [9] and are defined as follows (see p,q p,q also [10]): (u 1 , U1 ; λ1 : x), (u 2 , U2 ; λ2 ), · · · , (u p , U p ; λ p ) γ m,n γ m,n I p,q (z) = I p,q z (v1 , V1 ; μ1 ), · · · , (vq , Vq ; μq ) (u , U ; λ : x), (u j , U j ; λ j )2, p m,n = γ I p,q z 1 1 1 (v j , V j ; μ j )1,q 1 φ(s, x)z s ds, (1.1) = 2πi L and m,n I p,q (z)
(u , U
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