Some integrals involving k gamma and k digamma function

11 - The Gamma, Beta, Pi, and Psi Functions

This study presents a detailed investigation into the analytical properties of the generalized Gamma function introduced by Dilcher. We establish a fundamental recurrence relation and derive novel reflection formula for this generalized function, extending the classical identities known for the Euler Gamma function. Further, we analyze the generalized digamma function, the logarithmic derivative of the generalized Gamma function, and obtain a significant difference equation that characterizes its behavior. Explicit expressions for the evaluation of this generalized digamma function at half‐integer arguments are provided. The study also introduces the natural extension to generalized polygamma functions and reports on several special values for both classical and generalized cases. To complement the theoretical analysis, a graphical representation of the generalized digamma function is included, alongside supporting numerical computations that illustrate its properties and validate the derived results. Our findings provide a comprehensive framework that enhances the understanding of these generalized special functions and underscores their potential for broader application in mathematical physics and number theory.

We present some inequalities and a concavity property of the digamma function \psi=\Gamma'/\Gamma , where \Gamma denotes Euler's gamma function. In particular, we offer a new characterization of Euler's constant \gamma=0.57721\dots . We prove that -\gamma is the minimum of the harmonic mean of \psi(x) and \psi(1/x) for x>0 .

New approximations of the gamma function in terms of the digamma function

We propose a new sharp family of approximations which improve the results of Sandor and Debnath [On certain inequalities involving the constant e and their applications, J. Math. Anal. Appl. 249 (2000), pp. 569–582], Batir [Sharp inequalities for factorial n, Proyecciones 27 (2008), pp. 97–102] and other recent results. Then, we prove the complete monotonicity of a function involving the gamma function to establish new sharp upper and lower bounds for the gamma and the digamma functions.

In this paper, by deriving an inequality involving the generating function of the Bernoulli numbers, the author introduces a new ratio of finitely many gamma functions, finds complete monotonicity of the second logarithmic derivative of the ratio, and simply reviews the complete monotonicity of several linear combinations of finitely many digamma or trigamma functions.

This work explores the Julia and Mandelbrot sets of the Gamma function by extending the function to the entire complex plane through analytic continuation and functional equations. Various Julia and Mandelbrot sets associated with the Gamma function are generated using the iterative function , with different parameter values. To produce an accurate result using the integral definition of the Gamma function, a large number of terms would have to be added during the numerical integration procedure; this makes computation of Gamma function a very difficult task. To overcome this challenge, the Lanczos approximation of the Gamma function which presents an efficient and easy way to compute algorithms for approximating the Gamma function to an arbitrary precision is used. The resulting images reveal that the fractal (chaotic) behaviour found in elementary functions is also found in the Gamma function. The chaotic nature of the Julia and Mandelbrot sets provides a way of understanding complexity in systems as well as just in shapes.

In this paper, we mainly give much simple proof to Theorem 1.1 and (1.6) of Theorem 1.2 posed in the paper ?F. Qi, Limit formulas for ratios between derivatives of the gamma and digamma functions at their singularities, Filomat 27 (2013) 601-604.?

On some properties of the gamma function

In this paper, we consider multiplication formulas and their inversion formulas for Hurwitz–Lerch zeta functions. Inversion formulas give simple proofs of known results, and also show generalizations of those results. Next, we give a generalization of digamma and gamma functions in terms of Hurwitz–Lerch zeta functions, and consider its properties. In all the sections, various results are always proved by multiplication and inversion formulas.

Let varGamma (x) denote the classical Euler gamma function. The logarithmic derivative psi (x)=[ln varGamma (x)]'=frac{varGamma '(x)}{ varGamma (x)}, psi '(x), and psi ''(x) are, respectively, called the digamma, trigamma, and tetragamma functions. In the paper, the authors survey some results related to the function [psi '(x)]^{2}+ psi ''(x), its q-analogs, its variants, its divided difference forms, several ratios of gamma functions, and so on. These results include the origins, positivity, inequalities, generalizations, completely monotonic degrees, (logarithmically) complete monotonicity, necessary and sufficient conditions, equivalences to inequalities for sums, applications, and the like. Finally, the authors list several remarks and pose several open problems.

The purpose of this paper is to study gamma functions in the context of the theory of function fields of one variable over a finite field. In particular, we explore, among other things, their connection with the theory of Drinfeld modules of rank one (which is function field cyclotomic theory), their interpolations, functional equations, the nature of their special values (transcendence, algebraicity, connection with periods) and various analogies. Most of the facts about the classical gamma function are recalled when necessary. Readers may refer to [G-K] or Artin's book on the classical gamma function. Now we describe more precisely the contents of this paper. Let Fq be a finite field of characteristic p. Section 0 covers the background material on Drinfeld modules. It also describes the gamma function F for Fq[T] (with

We present some properties associated with the monotonicity and the complete monotonicity of the psi (or digamma) function and establish the higher-order estimate for the familiar Euler–Mascheroni constant γ. Relevant connections of the results presented here with those derived in earlier works are also pointed out.

The analytical performance evaluation of various coherent diversity combining reception techniques in the presence of multiuser interference is presented into this paper. Particularly, the outage probability of the generalized selection combining (GSC) and its two extreme cases, namely the maximum ratio combining (MRC) and the selection combining (SC), are evaluated. The multiple interferers undergo independent and non-identically distributed Nakagami-m fading channels, with distinct average received powers and arbitrary fading scales. New analytical outage expressions are presented including several channel fading conditions regarding the desired user's signal. Although the proposed approach is formed as an infinite series representation, it includes only simple elementary and Gamma functions, while it delivers a rapidly converging formula achieving a sufficiently high precision level. Moreover, the derived computational complexity of this function is thoroughly discussed, while a corresponding comparison with another popular technique indicates the usefulness of the proposed approach. Finally, the accuracy of the approach is verified with the aid of selected numerical results accompanied with equivalent simulation ones.

Using the series-expansion of digamma functions and other techniques, some monotonicity and logarithmical concavity involving the ratio of gamma function are obtained, which is to give a partially affirmative answer to an open problem posed by B.-N.Guo and F.Qi. Several inequalities for the geometric means of natural numbers are established.