Properties and Applications of a New Extended Gamma Function Involving Confluent Hypergeometric Function

Institute of Numerical Sciences, Kohat University of Science and Technology, Kohat 26000, Pakistan Department of Mathematics and Statistics, Hazara University, Mansehra, Pakistan, Pakistan Department of Mathematics, College of Arts and Sciences, Prince Sattam Bin Abdulaziz University, Wadi Aldawser 11991, Saudi Arabia Department of Mathematics and General Sciences, Prince Sultan University, P. O. Box 66833, Riyadh 11586, Saudi Arabia Department of Medical Research, China Medical University, Taichung 40402, Taiwan Department of Computer Science and Information Engineering, Asia University, Taichung 40402, Taiwan


Introduction
e classical gamma function was first found by Swiss mathematician Leonhard Euler (1707-1783) in his objective to generalize the factorial to nonintegral values. e classical gamma function and its related Pochhammer symbol are characterized as follows: and (z) ρ � Γ(z + ρ) Γ(z) , z(z + 1) . . . (z + n − 1), ρ � n ∈ N 0 , z ∈ C, e gamma function belongs to category of the special transcendental functions. It has applications in different fields, e.g., definite integration, asymptotic series, hypergeometric series, etc. In the last three decades, various extended forms of classical gamma functions are introduced by different researchers and mathematicians. Among those, some of the extended gamma functions and their related Pochhammer symbols are defined below.
Chaudhry and Zubair [1] introduced an extended gamma function, defined by Srivastava et al. [2] proposed the following extended form of Pochhammer symbol: Srivastava et al. [3] introduced an extended gamma function involving modified Bessel function and its associated Pochhammer symbol, defined as follows: and where R(κ) > 0, R(z) > 0 and K ] (.) is the modified Bessel function of order v defined by Safdar et al. [4] defined an extension of classical gamma function and its associated Pochhammer symbol involving Mittag-Leffler function, defined by and where R(ρ) > 0, R(z) > 0, and E σ (z) is the Mittag-Leffler function defined by Srivastava et al. [5] present the familiar incomplete gamma functions c (s, x) and Γ (s, x), and they introduce the incomplete Pochhammer symbols that led us to a natural generalization and decomposition of a class of hypergeometric and other related functions which are potentially useful in closed-form representations of definite and semiinfinite integrals of various special functions. Applications of these functions can be found in communication theory, probability distributions, and groundwater pumping modelling.
Şahin et al. [6] introduce a new generalization of the Pochhammer symbol by means of the generalization of extended gamma function as follows: Using the generalization of Pochhammer symbol, they give a generalization of the extended hypergeometric functions of one or several variables. Also, they obtain various integral representations, derivative formulas, and certain properties of these functions. e rest of the paper is organized as follows. In Section 2, we define a confluent hypergeometric gamma (CHG) function and derive its closed form in terms of Meijer's G-function, which is built in function of Computational Package Mathematica. In Section 3, a confluent hypergeometric Pochhammer (CHP) symbol is defined and some of its associate properties are also derived. In Section 4, we define an extension of generalized hypergeometric gamma (GHG) function and derive some of its associate properties. In Section 5, we obtain families of generating function relations. In Section 6, we present comparison of confluent hypergeometric gamma (CHG) function with classical and extended gamma functions for certain numerical values using Table 1. Comparison of integral and closed form of confluent hypergeometric gamma (CHG) function is shown in Table 2. Concluding remarks are given in Section 7.

Theorem 1. One has the following closed form for
where Proof. From [Rainville [7], p. 102, eorem 36], (12) becomes where k v is Macdonald function [9], defined by and I v (x) is a modified Bessel function of first kind [10], defined by  Using (15) and (16) and the residue theorem [11], one has which can be reexpressed as where R(p) > 0 and R(b), R(m), R(z) ≠ 0, − 1, − 2, . . . □ Remark 1. It is also important to mention here that in most of the extended cases, no closed form is provided but only new parameter in its function is inducted in the integrand of the integral of classical gamma function or its any extended gamma forms. Moreover, the proposed CHG function may find applications in statistics and a few associations with other special functions and polynomials.
Remark 2. Now, let m � b and α � 0; then, (12) reduces to extended gamma function defined by Zubair and Chaudhry [12], which is defined as If we consider p � 0, then (19) becomes the classical gamma function (1).

Incomplete Confluent Hypergeometric Gamma Functions.
Incomplete CHG functions are defined by and Now if we consider m � b and α � 0, then (20) and (21) are reduced to incomplete gamma function defined by Chaudhry and Zubair [12].

A Confluent Hypergeometric
Pochhammer Symbol e classical Pochhammer symbol is defined by Pochhammer symbol introduced by Srivastava et al. [2] is given by We propose a CHP symbol in following form: where Γ p (z; m, b, α) is defined in (12).
Proof. Using integral form of confluent hypergeometric function, (12) becomes Apply a one-one transformation but without including the boundaries. Let ] � ut, μ � t in the above equality and the Jacobian is J � 1/μ; then, one gets Interchanging the order of integration due to uniform convergence of the integrals and using some basic calculus, one obtains (28). □ Theorem 3. For product of two confluent hypergeometric gamma functions, one has the following result: Proof. Substituting t � σ 2 in (12), one gets Journal of Mathematics 5 erefore, Letting σ � c cos θ, ω � c sin θ in (33) gives (31). □ Remark 4. α � 0 in (33) gives the product of extended gamma function (see [13]).

Theorem 4.
For confluent hypergeometric gamma (CHG) function (12), one has the following recurrence relationship: Proof. e confluent hypergeometric functions , and 1 F 1 (ξ; τ + 1; z) in which one parameter is increased or decreased by unity are said to be contiguous to 1 F 1 (ξ; τ; z). e following relations exist between 1 F 1 and two of its contiguous functions [14].