Convexity properties and inequalities concerning the (p; k)-gamma function

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

In this paper, we introduce a new one-parameter deformation of the classical Gamma function, which we call a v-analogue. We also establish some properties generalizing those satisfied by the classical Gamma function. In addition, we establish some inequalities involving this new function.

In this short paper, we established q- analogues of some well-known inequalities for the classical Gamma function and some inequalities bounding the ratio Γ_q (x)/Γ_q (y), where Γ_q (.) is the q-analogue of the Gamma function. The method employed in presenting the results made use of strict logarithmic convexity of a function and classical mean value theorem involving the q-analogue of the psi function. At the end, the proven result provided the q-analogue of some known result concerning the classical Gamma function.

The objective of present research is to introduce and analyze a new special function that occurs as a part of the kernel of the integral representation of the λ-generalized Hurwitz-Lerch zeta functions. One important aspect of the analysis of special functions is to study their properties. It is found that λ-generalized gamma function satisfies the log convex and derivative properties. Recurrence relation and reflection formula are achieved that are always important to study the behavior of new functions. As an application, a new series representation of the λ-generalized Hurwitz-Lerch zeta functions is established. These new results are validated by analyzing their important cases that agree with the known results. It is interesting to note that the coefficients in the series representation of the family of zeta functions are generalized from “1” to “gamma function” and then to “generalized gamma” and “λ-generalized gamma functions” in a simple and natural way.

Convexity properties axe often useful in characterising and finding bounds for special function and their zeros, as well as in questions concerning the existence and uniqueness of zeros in certain intervals. In this survey paper, we describe some work related to the gamma function, the q-gamma function, Bessel and cylinder functions and the Hermite function.

The Euler–Maclaurin–Siegel and Abel–Plana summation formulae for the entire Riemann functional equation to handle the Riemann hypothesis

Finite fields analogs of the classical error function and incomplete gamma function are defined as complex functions over finite fields.

How to accurately process and achieve good transient performance in a short period of time is a key consideration factor for the system. A hyperbolic sine function is used to construct a novel predefined‐time convergent prescribed performance function. This algorithm introduces a set of new predefined standards for time convergence assessment based on gamma functions and Riemann zeta functions. By integrating performance indicators of speed, stability and efficiency into the design of the prescribed performance function, the performance framework ensures the achievement of establishing a comprehensive performance optimization model. The upper limit of the settling time is studied, and sufficient conditions for achieving predetermined time convergence are established, validated through experiments using unmanned surface vessels.

This paper is inspired by the work of J. S\'{a}ndor in 2006. In the paper, the authors establish some double inequalities involving the ratio $ \frac{\Gamma_{q}(x+1)}{ \Gamma_{q} \left( x+\frac{1}{2}\right)}$, where $\Gamma_{q}(x)$ is the $q$-deformation of the classical Gamma function denoted by $\Gamma(x)$. The method employed in presenting the results makes use of Jackson's $q$-integral representation of the $q$-deformed Gamma function. In addition, H\{o}lder's inequality for the $q$-integral, as well as some basic analytical techniques involving the $q$-analogue of the psi function are used. As a consequence, $q$-analogues of the classical Wendel's asymptotic relation are obtained. At the end, sharpness of the inequalities established in this paper is investigated.

In this paper, we prove some inequalities satisfied by the modified degenerate gamma function which was recently introduced. The tools employed include Holder’s inequality, mean value theorem, Hermite–Hadamard’s inequality, and Young’s inequality. By some parameter variations, the established results reduce to the corresponding results for the classical gamma function.

Recently, we have applied the generalized Littlewood theorem concerning contour integrals of the logarithm of the analytical function to find the sums over inverse powers of zeros for the incomplete gamma and Riemann zeta functions, polygamma functions, and elliptical functions. Here, the same theorem is applied to study such sums for the zeros of the Hurwitz zeta function ζ(s,z), including the sum over the inverse first power of its appropriately defined non-trivial zeros. We also study some related properties of the Hurwitz zeta function zeros. In particular, we show that, for any natural N and small real ε, when z tends to n = 0, −1, −2… we can find at least N zeros of ζ(s,z) in the ε neighborhood of 0 for sufficiently small |z+n|, as well as one simple zero tending to 1, etc.

On Müntz-type formulas related to the Riemann zeta function

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.

Some two-sided inequalities for multiple Gamma functions and related results

An algorithm for parametric elementary integration over differential fields constructed by a differentially transcendental extension is given. It extends current versions of Risch's algorithm to this setting and is based on some first ideas of Graham H. Campbell transferring his method to more formal grounds and making it parametric, which allows to compute relations among definite integrals. Apart from differentially transcendental functions, such as the gamma function or the zeta function, also unspecified functions and certain families of iterated integrals such as the polylogarithms can be modeled in such differential fields.