Unified integral associated with the generalized V-function

*Correspondence: s.k.q.alomari@fet.edu.jo; shridehalomari@bau.edu.jo 2Department of Physics and Applied Sciences, Al-Balqa Applied University, Amman, Jordan Full list of author information is available at the end of the article Abstract In this paper, we present two new unified integral formulas involving a generalized V-function. Some interesting special cases of the main results are also considered in the form of corollaries. Due to the general nature of the V-function, several results involving different special functions such as the exponential function, the Mittag-Leffler function, the Lommel function, the Struve function, the Wright generalized Bessel function, the Bessel function and the generalized hypergeometric function are obtained by specializing the parameters in the presented formulas. More results are also discussed in detail.


Introduction and preliminaries
Fractional calculus is as old as the conventional calculus and has been recently applied in various areas of engineering, science, finance, applied mathematics and bioengineering. The V-function is an important special function that provides solutions to a 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 addition, a number of researchers (see [10,11,13,17,19,22,29,30]) have studied in depth properties, applications and diverse extensions of a range of operators of fractional calculus, this field being very active and extensive around the world. One may refer to the research monographs [12] and [21] for further investigations in the area. Recently, the V-function is defined by Kumar [14] as follows: 2. p, q, and r are natural numbers, 3. a u , b v ≥ 1(u = 1, . . . , p; v = 1, . . . , q), 4. η > 0, (μ) > 0, (h) > 0, z is a complex variable and ξ is an arbitrary constant, 5. the series on the RHS of (1.1) converges absolutely if p < q or p = q with |l(z/2) ζ | ≤ 1. For further information on the constraints of the convergence of the RHS of the series (1.1), we refer to Refs. [15,16]. The V-function defined by (1.1) is of general character as it assimilates a variety of valuable functions such as the Macrobert E-function, the exponential function [4], the generalized Mittag-Leffler function [9,23,31,34], the Lommel function [8], the Struve function [26,33,35], the generalized Bessel function [27], the Bessel function [5,36], the generalized hypergeometric function [4,32,37,38] and the unified Riemann-Zeta function [7].

Main results
In this section, we establish four generalized integral formulas for the V-function. These formulas are given by the following theorems.
. Then the following integral holds true: Proof For the convenience of the reader, we denote the left-hand side of (2.1) by 1 . Therefore, by invoking (1.1) in the integrand (2.1) and interchanging the order of integration and summation, which is verified by the uniform convergence of the involved series under the given conditions, we get Hence, on applying the integral formula (1.13) for the integral in (2.2), we, under the valid conditions, obtain the following expression: which is the desired result.

(2.3)
Proof By following a technique similar to what has already been used in the proof of Theorem 2.1, we can easily prove the integral formula (2.3). Therefore, we omit the detailed proof.
. Then the following integral holds true: Proof For more convenience, we denote the left-hand side of (2.4) by 2 . Therefore, by invoking (1.1) in the integral part of (2.4) and interchanging the order of integration and summation, which is verified by uniform convergence of the involved series under the given conditions, we obtain Now, upon applying the integral formula (1.14) to the integral part of (2.5) we obtain the following expression under their valid conditions: which is the desired result.

Special cases
In this section, we aim to present some special cases by adopting certain advisable values of the parameters imposed on Theorems 2.1, 2.