A New Class of Extended Hypergeometric Functions Related to Fractional Integration and Transforms
The focus of this research is to use a new extended beta function and develop the extensions of Gauss hypergeometric functions and confluent hypergeometric function formulas that are presumed to be new. Four theorems have also been defined under the generalized fractional integral operators that provide an image formula for the extension of new Gauss hypergeometric functions and the extension of new confluent hypergeometric functions. Moreover, discussed are analogous statements in terms of the Weyl, Riemann–Liouville, Erdélyi–Kober, and Saigo fractional integral and derivative operator types. Here, we are also able to generate more image formulas by keeping some integral transforms on the obtained formulas.
1. Introduction and Preliminaries
A special function is any of a number of mathematical functions that emerge in the solution of many classical physics problems. A recurrent theme in these challenges is the flow of electromagnetic, acoustic, and thermal energy. For the specific kind of special role of scientists and engineers, it has become an essential resource now a day. And in many fields, such as physical science, mathematical science, and engineering, this feature plays a very important role. There are numerous ways to define special functions. A series or an adequate integral can be used to define several specific functions of a complex variable. Special functions, such as Bessel functions, Whittaker functions, Gauss hypergeometric functions, and Jacobi, Legendre, Laguerre, and Hermite polynomials, have been continuously developed. Polynomial sequences are very important in applied mathematics (see [1–4]). Fractional calculus is a noninteger order generalisation of classical differentiation and integration. Riemann–Liouville, ErdéIyi–Kober, Hadamard, Caputo, Hilfer, Liouville–Caputo, Grünwald–Letnikov, Riesz, Coimbra, and Weyl all provided important definitions of fractional derivatives.
Many substances’ consciousness and natural processes are characterised by fractional derivatives. As a generalisation of classical integer-order differential equations, fractional ordinary and partial differential equations exist. Fractional differential equations have been used to examine a variety of phenomena. It becomes more common to use it to illustrate problems in biology, electrodynamics, elasticity, viscoelasticity, fluid dynamics, physics, engineering, and a variety of other subjects (see [5–10]). Saigo  with generalized Gauss hypergeometric function (GHF) in the kernel and Saigo and Maeda  with Appell function in the kernel introduced generalized fractional integral operators.
In our analysis, we have to remember the subsequent pair of Saigo–Maeda fractional integral operators.
Let such that , , and the generalized fractional integral operator connecting Appell’s function or Horn’s function with the kernel  is defined as follows:
The Appell’s function or Horn’s function in two variables  is defined as
Remark 1. The Appell’s function diminishes into Gauss hypergeometric function by the subsequent relationEquations (1) and (2) reduce to Saigo operators, which are generalized fractional integral operators bound to the Gauss hypergeometric function if setting and reduces it to the Saigo type, Erdélyi–Kober, Riemann–Liouville (R-L), and Weyl type fractional integral operator, respectively, as follows:where .
The operator contains Saigo type, Riemann–Liouville (R-L), and Erdélyi–Kober fractional integral operator, and combines the Saigo type, Weyl, and Erdélyi–Kober fractional integral operator defined by Saigo .(i), and (ii) and
Lemma 1. Suppose and be s.t. , then the subsequent formula is validwhere .
Lemma 2. Suppose be such that ; then, we have the subsequent relationwhere
Lemma 3. Suppose be such that ; then, we have the subsequent relation
Lemma 4. Let , then we have the subsequent relation
Suppose and are the two series whose convergence radii are determined by and , respectively. Then, the series described by their Hadamard product  iswhere the radii of convergence satisfy .
If one of the series specifies it and the radii of convergence of the other are greater than 0, the Hadamard series describes the complete function. Using the Hadamard product, we can deconstruct a developing function into two famous functions. For instance, can be crumbled as follows:
2. A Class of Extended Hypergeometric Functions
For the present investigation, we use the extension of the hypergeometric function specified by Palsaniya et al.  in the following way:where extended beta function is defined as
If and , then equation (18) reduces to the classical beta function.
Throughout this part, we use the latest extension of the generalized Gauss hypergeometric function (GHF) and the confluent hypergeometric function . If we take in equations (16)–(18), the resultant definition was used by Chand et al. . In the field of special functions, many kinds of extensions of Gamma and Beta functions are done by many researchers (see [17–24]).
3. Fractional Integration of the New Extended Hypergeometric Function
In this segment, we evaluate several fractional integral formulas for the generalized hypergeometric function and confluent hypergeometric function using the concept of the Hadamard product .
Theorem 1. Let and be such that ,
Proof. Let us consider that the left-hand side of equation (20) is symbolized by . In equation (16), if we reverse the order of integration and summation, we getUsing the result which is defined in (8), we haveAfter simplification, using the property of the Pochhammer symbol , we obtainFinally, using the concept of the Hadamard product in the above equation, then it will change into the required result (20).
Theorem 2. Suppose , be such that , , then
Proof. Using the formula in equation (16), we designate the left side of equation (24) as and then exchange the order of integration and summation, yielding the following result:Using the result of Lemma 1 which is defined in equation (9), after some simplification, we obtainUsing the concept of Hadamard product, we get the desired result.
Theorem 3. Let and be such that , such that , then
Theorem 4. Let and be such that , such that , then
Corollary 1. Suppose and be s.t. and , then
Corollary 2. Let and be such that , then
Corollary 3. Let and be such that and and , then
Corollary 4. Let , be such that and and , then
Remark 5. If we choose and in equations (16) and (17), the new extension of Gauss hypergeometric function and new extension of confluent hypergeometric function convert into classical function, and all results moreover reduce into fractional integration of classical functions in terms of Saigo–Maeda, Saigo, Erdélyi–Kober, Riemann–Liouville, etc.
4. Integral Transforms of Extended Gauss and Confluent Hypergeometric Functions
In this section, we demonstrate several theorems and corollaries by using different kinds of transforms like Beta transform and Laplace transform based on the results acquired in the previous section.
4.1. Beta Transform
The Beta transform of is demarcated as
Theorem 5. Suppose and be such that and , then
Proof. For convenience, we refer to the left-hand side of (34) by ; then applying the definition of beta transform, we haveand we may now use the formula in (16) and change the order of integration and summation to get the following result:By applying the formula given in (8) and using the definition of beta transform in the above equation, then it will reduce toFurther simplification of (37) yieldsInterpreting (38) with the help of (16), we obtainNow, interpreting (39) with the view of the concept of the Hadamard product, we have the required result given in equation (34).
Theorem 6. Let , and , , then