ON A GENERALIZED GAMMA FUNCTION AND ITS PROPERTIES

unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract. In this work, we introduce a new generalized gamma function and establish its validity through the Bohr-Mullerup theorem. We also establish the generalized Euler reflection formula and some other properties related to the generalized gamma function.The concept of powers of logarithm was largely used to establish the results.


INTRODUCTION
Euler introduced the gamma function with the goal to generalize the factorial to non integer values. Some prominent mathematicians such as Gauss, Legendre, Weierstrass, among others also studied it. The Gamma function belongs to the class of special functions and some mathematical constants such as the Euler-Mascheroni constant occur in its study.
In studying the Riemann zeta function and other special functions, the gamma function plays a vital role and has been the subject of study for over 300 years. The gamma function is still being studied by contemporary mathematicians and yet there seems to be so much to study about it.
The gamma function is essential for modeling situations involving continuous change and has applications in calculus, differential equations, statistics, fluid mechanics, quantum physics and complex analysis.
A generalized gamma function Γ k (z) for k ∈ N 0 was introduced in [1] which connects to the constant γ k as Γ(z) does to γ. Motivated by a series form of the generalized Euler-Mascheroni constants and through the concept of powers of logarithms, some properties of the gamma function were presented in [1].
The aim of this paper is to establish another form of a generalized gamma function and its properties.

PRELIMINARIES
The gamma function, a generalization of the factorial to non-integer values, was defined by Euler as Gauss rewrote Euler's product representation of the gamma function as The integral representation of the gamma function was also defined by Euler as Weierstrass also established another product representation of the gamma function as where γ is the Euler Mascheroni constant.
Eventhough Euler pioneered the theory of complex analysis, he did not consider the gamma function of a complex argument. Gauss did by establishing its multiplication theorem. Karl Weierstrass also introduced the product representation as given by (2.4).
In [3] a wide class of generalizations for the gamma function was studied and a special case for this class of generalizations was also studied by Dilcher [1]. In particular, the generalized gamma function Γ k (z) was introduced for k ∈ N 0 and some basic properties such as product and series expansions of a generalized gamma function were developed in [1]. He also established a series expansion for the generalized Euler constant for k ∈ N.
It was observed in [4] that an assymptotic expansion of Dilcher's generalized gamma function converges for k = 1 and the closed form was unknown for k > 1.
Some results in [1] are connected with those in [2] and a question was posed in [2] whether it is possible to extend the gamma function by analytic continuation. This study establishes a new generalized gamma function and its properties.
The generalized Euler-Mascheroni constants are defined as ln k j j , k = 0, 1, 2, ... (2.5) and are coefficients of the Laurent expansion of the Riemann zeta function ζ (s) about s = 1: where A k = (−1) k k! γ k . The constants A k were first defined by Stieltjes in 1885 and have been studied by other authors.
It is worth noting that γ 0 = γ is the Euler constant and is closely related to the gamma function.
Observe that if s = 0, the above Laurent expansion gives Thus, the real part of the nontrivial zeroes of the Riemann zeta function is associated with the generalized Euler-Mascheroni constants.
The Euler's reflection formula is given by The Riemann zeta function is defined by and its derivatives are given by The stirling numbers of the first kind, s(m, j), is defined by the generating function as where |z| < 1.
Alternatively, stirling numbers of the first kind are also defined by From (2.9), we see that Dilcher defined the gamma function as [1] (2.12) This definition excluded non positive real numbers in its domain. That means it is not The functional equation was obtained as The Weierstrass form of the generalized gamma function was also established as (2.14) 1 It was also discovered in [1] that for |z| < 1, and a generalized Euler reflection formula given as In particular, s 0 = sin(πz) πz . A consequence of (2.16) yields (Dilcher, 1994) Let z ∈ D be fixed and k ∈ N. The identity holds as n → ∞.

MAIN RESULTS
We begin this section by presenting a new generalized gamma function which is pivotal in achieving further results of this paper.
A new generalization of the gamma function is introduced as follows: We check the validity of   (c) Taking logarithm on both sides of (3.1), we obtain Differentiating, we get For z ∈ R + , we have ln Γ k (z) ≥ 0, (3.6) and the proof is complete. Now, we present the following theorem which establishes the equality of equations (2.12) and (3.1) for the same domain.
Theorem 3.5. Let k ∈ N 0 and z ∈ C \ R − . Then This simplifies to Alternatively, from (2.12), we have, Hence, Theorem 3.6. Let z ∈ D = C \ (Z − ∪ 0). Then the identity Proof. From (3.1) and using j = ∏ n j=1 1 + 1 j , we obtain Taking logarithm on both sides of (3.9) and applying Lemma 2.1 gives By introducing convergence factors we obtain where γ k is the generalized gamma function.
This completes the proof.
By further taking logarithm on both sides of (3.14) gives Proof. Replacing z by −z in (3.2) and multiplying the result by Γ k (z), we have Simplifying further yields This completes the proof of the first part of the theorem.
where H m−1 is the (m-1)th harmonic number.