Research Article Composition Formula for Saigo Fractional Integral Operator Associated with V-Function

In this study, we form integral formulas for Saigo’s hypergeometric integral operator involving V-function. Corresponding assertions for the classical Riemann–Liouville (R-L) and Erd´elyi–Kober (E-K) fractional integral operator are extrapolated. Also, by putting in the transformations of Beta and Laplace, we can establish their composition formulas. By selecting the appropriate parameter values, the V-function may be reduced to a variety of functions, including the exponential function, Mittag–Leffler, Lommel, Struve, Wright’s generalized Bessel function, and Bessel and generalized hypergeometric function.


Introduction and Preliminaries
Calculus with fractional orders is a branch of mathematics that develops from typical de nitions of calculus with integer orders of integral and derivative operators, just how fractional exponents develop from integer exponents. In recent years, fractional calculus has been used in di erent elds of technology, research, plasma physics, economics, nonlinear control theory, applied maths, and bio-engineering. e V-function plays an important role to develop solutions to critical problem in terms of fractional-order integral and di erential equations. e computations of fractional integrals and fractional derivatives involving transcendental functions of one and several variables are important because of the usefulness of their results, e.g., for evaluating di erential and integral equations. e characteristics, execution, and various extensions of a number of fractional calculus operators have studied in depth by researchers (see [1][2][3][4][5][6]). We recollect the Saigo fractional integral operator containing Gauss's hypergeometric function 2 F 1 (.) as a kernel. Let l, m, ξ ∈ C and x > 0; then, the generalized fractional integration operators related with Gauss hypergeometric function due to Saigo [7,8] are de ned as follows: If we set m � 0 in equations (1) and (3), we get the E-L fractional integral operators identified as e operators I l,m,ξ 0,x and I l,m,ξ x,∞ include, as their special case, m � −l, the fractional integral operator of (R-L) and Weyl type as For the present study, power function formulas of the fractional integral operators discussed above are needed as given in the following lemmas. Lemma 1. Let x > 0 and l, m, ξ, ∈ C be such that R(l) > 0; then, there exists the relation Lemma 2. Let x > 0 and l, ξ, ζ ∈ C be such that R(l) > 0; then, there exists the relation Lemma 3. Let x > 0 and l, ζ ∈ C be such that R(l) > 0; then, there exists the relation: Kumar [9] recently defined the V-function as follows: where . . , q), and g u (u � 1, . . . , r) are real numbers (2) p, q, and r are natural numbers and λ is an arbitrary constant (5) e series on the RHS of (11) converges absolutely if q > p or q � p with |L(z/2) k | ≤ 1 For descriptions of the series' convergence constraints on (11) RHS, simply review [10][11][12]. e V-function defined by (11) is of a general character since it assimilates and applies a variety of valuable functions such as Macrobert's E-function and exponential function [13], generalized Mittag-Leffler function [14][15][16][17][18], Lommel's function, Struve's function, generalized Bessel function and Bessel function [19][20][21][22][23][24], generalized hypergeometric function [13,25,26], and unified Riemann-zeta function [27]. e purpose of this work is to evaluate the compositions of the generalized fractional integration operators (1) and (3) with the (11) V -function. Additionally, equivalent claims for the classical R-L and E-K fractional integral operators are evaluated. e conclusions mentioned in conjunction with the corollaries in theorems are undoubtedly innovative and conceptually valuable. Additionally, we derive the composition formulas for the findings stated in theorems using the Beta and Laplace transforms. Finally, as previously said, establish them as special instances on the primary result in conjunction with various special functions, the hypergeometric function, and so on.

Fractional Integration of V-Function
In this section, we develop image formulas for the V-function involving the Saigo fractional integral operator's left and right sides. e following theorems give these formulas.
en, the subsequent Saigo hypergeometric fractional integral I l,m,ξ 0,x of V-function holds true: Proof. For convenience, using definitions (1) and (11), we denote the L. H. S. of the result (12) by I1; by changing the order of summation and integration, we obtain By applying relation (7) in (13), we get R.H.S. of (12). □ en, the subsequent Saigo hypergeometric fractional integral I l,m,ξ x,∞ of V-function holds true: Proof. To derive (14), we denote (14) by I 2 to L.H.S.; using definitions (3) and (11) and by changing the order of summation and integration, we obtain

Beta Transform of the Composition Formulas (12) and (14)
In this section, we develop some theorem that includes the results obtained in the previous section concerning the integral transformation.

Journal of Mathematics
Proof. e proof of eorem 4 is similar manner of eorem 3.