Related papers: An algorithm for evaluating the Gamma function and…
An inequality concerning ratios of gamma functions is proved. This answers a question of Guo and Qi (2003).
Observing a multiple version of the divisor function we introduce a new zeta function which we call a multiple finite Riemann zeta function. We utilize some $q$-series identity for proving the zeta function has an Euler product and then,…
In this article we derive, using the Lagrange inversion theorem and applying twice the Fa\`a di Bruno formula, an expression of the minimum of the Gamma function $\Gamma$ as an expansion in powers of the Euler-Mascheroni constant $\gamma$.…
We introduce a family of regularized functionals $g_n(x)$ that generalize the Euler--Mascheroni constant $\gamma$. They arise from a weighted regularization of Clausen-type trigonometric sums, and admit explicit integral representations,…
The Riemann Hypothesis has been of central interest to mathematicians for a long time and many unsuccessful attempts have been made to either prove or disprove it. Since the Riemann zeta function is defined as a sum of the infinite number…
The gamma model is a generalized linear model for gamma-distributed outcomes. The model is widely applied in psychology, ecology or medicine. In this paper we focus on gamma models having a linear predictor without intercept. For a specific…
The definition of conservative-irreversible functions is extended to smooth manifolds. The local representation of these functions is studied and reveals that not each conservative-irreversible function is given by the weighted product of…
We show that an apparently overlooked result of Euler from \cite{E421} is essentially equivalent to the general multiplication formula for the $\Gamma$-function that was proven by Gauss in \cite{Ga28}.
Algorithms for computing rational generating functions of solutions of one-dimensional difference equations are well-known and easy to implement. We propose an algorithm for computing rational generating functions of solutions of…
We establish sharp inequalities involving the incomplete Beta and Gamma functions. These inequalities arise in the approximation of generalized Bernstein functions by higher order Thorin-Bernstein functions. Furthermore, new properties of a…
We construct a complex entire function with arbitrary number of variables which has the following property: The infinite set consisting of all the values of all its partial derivatives of any orders at all algebraic points, including zero…
We use well-known limit theorems in probability theory to derive a Wallis-type product formula for the gamma function. Our result immediately provides a probabilistic proof of Wallis's product formula for $\pi$, as well as the duplication…
A new general and unified method of summation, which is both regular and consistent, is invented. It is based on the idea concerning a way of integers reordering. The resulting theory includes a number of explicit and closed form summation…
Quantum computing is a new model of computation, based on quantum physics. Quantum computers can be exponentially faster than conventional computers for problems such as factoring. Besides full-scale quantum computers, more restricted…
An asymptotic expansion of a ratio of products of gamma functions is derived. It generalizes a formula which was stated by Dingle, first proved by Paris, and recently reconsidered by Olver.
The primary goal of this paper is to introduce and investigate generalized incomplete exponential functions with matrix parameters. Integral representation, differential formula, addition formula, multiplication formula, and recurrence…
I present a novel mathematical technique for dealing with the infinities arising from divergent sums and integrals. It assigns them fine-grained infinite values from the set of hyperreal numbers in a manner that refines the standard…
In the paper, we establish an inequality involving the gamma and digamma functions and use it to prove the negativity and monotonicity of a function involving the gamma and digamma functions.
We consider the functional inverse of the Gamma function in the complex plane, where it is multi-valued, and define a set of suitable branches by proposing a natural extension from the real case.
We give extensions of results on nonnegative matrix semigroups which deduce finiteness or boundedness of such semigroups from the corresponding local properties, e.g., from finiteness or boundedness of values of certain linear functionals…