Certain fractional integral operators pertaining to S-function

Abstract Fractional integral operators introduced by Saxena and Kumbhat involving Fox’s H-function as kernel are applied, and find new image formulas of S-function and properties are established. Also, by implementing Euler, Whittaker and K-transforms on the resulting formulas. On account of S-function, a number of results involving special functions can be obtained merely by giving particular values for the parameters.


Introduction and preliminaries
In recent years, the fractional calculus has become a significant instrument for the modeling analysis and assumed a significant role in different fields, for example, material science, science, mechanics, power, economy and control theory (see details: Alaria et al. (2019), Berdnikov and Lokhin (2019), Drapaca (2018), and Hammachukiattikul (2019)). In addition, a number of researchers like (Agarwal and Jain (2011), Baleanu (2009), Kalla (1969), and Kilbas (2005)) have studied indepth level of properties, applications and diverse extensions of a range of operators of fractional calculus. Also, on other analogous topics is very active and extensive around the world. One may refer to the research monographs Kiryakova (1994) and Miller and Ross (1993), and the recent papers Kilbas et al. (2006), Mathai et al. (2010), Samko et al. (1993), and Suthar and Amsalu (2019). Recently, Saxena and Daiya (2015) defined and study a special function called as S-function, its relation with other special functions, which is a generalization of k-Mittag-Leffler function,

PUBLIC INTEREST STATEMENT
The fractional calculus functions are very useful almost in all areas of applied Mathematics that provides solutions to the number of problems formulated in terms of fractional order differential, integral and difference equations; therefore, it has recently become a subject of interest for many authors in the field of fractional calculus and its applications in astrophysics, biosignal processing, fluid dynamics, non-linear control theory and stochastic dynamical system, and so on. In this paper, we have evaluated two theorems for generalized fractional integral operators involving Fox's H-function as kernel, applied on the S-function and also point out their relevance properties and the known results.
Some important special cases of S-function are enumerated below: Saxena et al. (2014): (ii) Again, for k ¼ τ ¼ 1; the S-function is the generalized K-function, defined by Sharma (2011): Now, we recall the Saxena and Kumbhat (1974) operators involving Fox's H-function as kernel, by means of the following equations: where U and V represent the expressions and respectively with �; υ>0: The sufficient conditions of above said operators are given below: (3) f ðxÞ 2 L P 0; 1 ð Þ; (4) arg λ j j< θπ 2 ; θ>0; where, Fox H-function Fox (1961), in operator (1.4) For the convergence conditions together with the conditions of analytical continuations of Hfunction, one can see Mathai and Saxena (1978) and Mathai et al. (2010). Throughout this paper, we assume that the above conditions are fulfilled by the said function.
The Euler transform (Sneddon (1979)) of a function f ðzÞ is defined as: Due to Whittaker transform (Whittaker and Watson (1996)), the following result true: The following integral equation defined in term of K-transform (Erdélyi et al. (1954)) as: where < h ð Þ>0; K υ x ð Þ is the Bessel function of the second kind defined by (Srivastava et al. (1982), p. 332) where W 0;υ : ð Þ is the Whittaker function defined in equation (1.9).
The above result given in (Mathai et al. (2010), pp. 54, Eq. 2.37) will be used in evaluating the integrals.
In see of the effectiveness and extraordinary significance of the fractional integral operators given by Saxena and Kumbhat in specific issues, the authors establish the image formulas and derive certain properties of S-function. The results obtained here involve special functions like 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.

Images of S-function under the fractional integral operators
In this part, we obtain the images of S-function under the generalized fractional integral operators defined in 1.4 and 1.5. Theorem 1. Let ρ; δ; ω; τ 2 C; <ðρÞ>0; < # ð Þ>0; k 2 <; <ðρÞ>k<ðτÞ; x>0; the fractional integration R η;σ x;γ of S-function exists, under the condition c À 1 þ d À 1 ¼ 1; . . . ; mÞ: Then there holds the result: Proof. Assume } be the left-hand side of (2.5), using (1.1) and (1.5), we have Changing the order of the integration under the valid condition provided in the theorem statement, we get Let the substitution x γ =t γ ¼ u; then t ¼ x=u 1=γ ð Þ in above term and applying beta function, we get Interpreting the right-hand side of (2.6), in view of the definition (1.2), (1.3) and (1.6), we arrive at the result (2.5).

Special cases
(i) If we put p ¼ q ¼ 0, in Theorem 1 and Theorem 2, then we find the following interesting results on the right which are known as generalized k-Mittag-Leffler function.
Corollary 1. Eq. (2.1) reduces in the following form: Corollary 2. Eq. (2.5) reduces in the following form: (ii) For putting k ¼ τ ¼ 1 in Theorem 1 and Theorem 2, then we get the following interesting results on the right is known as K-function.
Corollary 3. Eq.(2.1) reduces in the following form: Corollary 4. Eq.(2.5) reduces in the following form: Suthar, Cogent Mathematics & Statistics (2020) (iii) For taking τ ¼ k ¼ ω ¼ 1 in Theorem 1 and Theorem 2, then we obtain the following results on the right is known as M-series.

Integral transforms of S-function involving fractional integral operators
In this part, the results established in Theorems 1 and Theorems 2 have been obtained in terms of Euler, Whittaker and K-transforms.
Theorem 7. Follow stated Theorem 1 for conditions on parameters, with Then the subsequent result true: Proof. Using (2.1) and (1.10), it gives we get Suthar, Cogent Mathematics & Statistics (2020) simplification on right-hand side of (4.7), we obtain at the result (4.6).

Properties of integral operators
Here, we established some properties of the operators concerning with Theorem 1 and Theorem 2. These properties are given in the compositions of power function.

Conclusions
In the present paper, we have studied the properties of S-function under the extension of generalized fractional integral operators given by Saxena 2018)). Also, the special functions involved here can be reduced in simpler functions, those have a variety of applications in different domains of science and technology and can be observed as special cases, those we have not mentioned here explicitly.