Euler-Type Integral Operator Involving S-Function

The object of the present paper is to establish some interested theorems on Euler-type integrals involving S-function, which is defined by Saxena and Daiya. Further, we reduce some special cases involving various known functions like the k-Mittag-Leffler function, K-function, and M-series.


Introduction
In recent years, fractional calculus has become a significant instrument for modeling analysis and has assumed a significant role in different fields, for example, material science, science, mechanics, power, economy, and control theory. Additionally, a variety of researchers researched a selection of fractional calculus operators in-depth with a scope on properties, implementations, and complex extensions. Also, other analogous topics are very active and extensive around the world. Recently, Saxena and Daiya [1] defined and studied a special function called an S-function and its relation with other special functions, which is a generalization of K-function, M-series, k-Mittag-Leffler function, Mittag-Leffler-type functions, and many other special functions. For a detailed account of the S-function along with its properties and applications, one can read [2][3][4][5]. Motivated by these research findings, we tend to establish some of the theorems of concern for the Euler form integral concerning S-function and its related special function. Such specific functions have recently been established as important applications for solving problems in biological sciences, genetics, physics, and engineering.
Several major special cases of the S-function are described below: (i) For p = q = 0, the generalized k-Mittag-Leffler function is from Saxena et al. [9] (see [10,11]).
Now, we mention the basic beta function indicated by Bða, bÞ that is described by Euler's integral [17] as Euler has extended the factorial function from the natural number domain to the gamma function: defined over the right half of the complex plane. Chaudhry and Zubair [18] expanded the scope of these functions to the entire complex plane by adding an exp ð−A/uÞ regularization element in the integrand of equation (10). With A > 0, this element explicitly excludes the singularity resulting from the u = 0 limit. For A = 0, this element is unity, and we get the gamma function originally used. We mention the relation below ( [19], p. 20 (2)).
where K a ðxÞ is the altered Bessel function of the second kind of order n. We consider Riemann's zeta function ζðxÞ described by the series ( [20], p. 102 (2.101)) is useful for comparison testing to provide convergence or divergence of certain series. Zeta function is directly connected to the gamma function logarithm and the polygamma functions. The exp ð−A/uÞ regularizer has also proven to be very useful in expanding the zeta function of Riemann's domain, thereby supplying connections that could not have been achieved with the original zeta function. Considering the usefulness of the above regularizer for gamma and zeta functions, Chaudhry et al. [19] proposed the following expansion of Euler 's beta function as follows: Lee et al. [21] presented the extended Euler beta functions in the continuation of his research and described it as In addition, new Euler generalizations of k-beta functions are described by Khan et al. [22] as follows: where k ∈ R, ðRðpÞ > RðmÞ > 0Þ:
In this article, several Euler-type integral operator theorems concerning S-function were obtained and some specific cases were addressed.

Special Cases
In this section, we define as special cases of our key results the following potentially useful integral operators that include generalized k-Beta type functions and generalized Beta type functions: (1) On setting p = q = 0 in Theorem 1, we get where E ε,τ k,σ,η ðzÞ is a k-Mittag-Leffler function (5) (2) On setting k = τ = 1 in Theorem 1, we get where K σ,η,ε ðp,qÞ ðzÞ is a K-function (6)