Pathway Fractional Integral Formulas Involving S-Function in the Kernel

In recent years, fractional calculus has become a significant instrument for the modeling analysis and plays a significant role in different fields, for example, material science, science, mechanics, power, economy, and control theory. In addition, a number of researchers have investigated a variety of fractional calculus operators in the depth level of properties, implementation methods, and complex modifications. Other analogous topics are also very active and extensive around the world. One may refer to the research monographs in [1, 2]. S-function. Recently, Saxena and Daiya [3] defined and studied a special function called as S-function (also see [4]) and its relation with other special functions, which include generalized K-function, M-series, k-Mittag–Leffler function, Mittag–Leffler type functions, and other many special functions. ,ese special functions have recently found essential applications in solving problems in applied sciences, biology, physics, and engineering. ,e S-function is defined for σ, η, ε, τ ∈ C, R(σ)> 0, k ∈ R, R(σ)> kR(τ), li(i � 1, 2, 3, . . . , p), mj(j � 1, 2, 3, . . . , q), and p< q + 1 as


Introduction and Preliminaries
In recent years, fractional calculus has become a significant instrument for the modeling analysis and plays a significant role in different fields, for example, material science, science, mechanics, power, economy, and control theory. In addition, a number of researchers have investigated a variety of fractional calculus operators in the depth level of properties, implementation methods, and complex modifications. Other analogous topics are also very active and extensive around the world. One may refer to the research monographs in [1,2].
e pathway model for a real scalar ς and scalar random variables is represented by the probability density function (p.d.f.) in the following manner: ; v >0 and ς and c denote the pathway parameter and normalizing constant, respectively. Additionally, for ς ∈ R, the normalizing constants are expressed in the following way: It is noted that if ς < 1, finite range density with (8) can be considered a member of the extended generalized type-1 beta family. Also, the triangular density, the uniform density, the extended type-1 beta density and various other probability density functions are precise special cases of the pathway density function defined in (8) for ς < 1.

Pathway Fractional Integral Operator of S-Function
In this section, we establish the PFI formula involving the S-function which is stated in eorems 1 and 2.
en, the following formula holds true: Proof. We indicate the RHS of equation (13) by I 1 , and invoking equations (1) and (7), we have Now changing the order of integration and summation, we obtain Using the substitution u � a(1 − ς)ζ/x, we can change the limit of integration into the following: Now, by calculating the inner integral and using the beta function formula, we obtain the following: Using (3), we obtain Mathematical Problems in Engineering Once again, using (3), we obtain which gives the required proof of eorem 1.
□ Corollary 1. If we put p � q � 0, then (13) leads to the subsequent result of generalized k-Mittag-Leffler function: Proof. We consider (4) and p � q � 0 in eorem 1, and we obtain the desired result in (13).

Concluding Remarks
In the present paper, we have established two pathway fractional integral formulae associated with the more generalized special function called as S-function. e results Mathematical Problems in Engineering obtained here involve special functions such as k-Mittag-Leffler function, K-function, and M-series, due to their general nature and usefulness in the theory of integral operators and relevant part of computational mathematics. Also, the special functions involved here can be reduced to simpler functions, which have a number of applications in various fields of science and technology and can be found as special cases that we have not specifically stated here.

Data Availability
No data were used to support this study.