Related papers: The generalized-Euler-constant function $\gamma(z)…
We present two integral representations of the logarithm of the Glaisher-Kinkelin constant. The calculations are based on definite integral expressions of $\log\Gamma(x)$, $\Gamma$ being the usual Gamma function, due respectively to F\'eaux…
We derive an integral expression $G(z)$ for the reciprocal gamma function, $1/\Gamma(z)=G(z)/\pi$, that is valid for all $z\in\mathbb{C}$, without the need for analytic continuation. The same integral avoids the singularities of the gamma…
Euler's Gamma function $\Gamma$ either increases or decreases on intervals between two consequtive critical points. The inverse of $\Gamma$ on intervals of increase is shown to have an extension to a Pick-function and similar results are…
We study a problem of finding good approximations to Euler's constant $\gamma=\lim_{n\to\infty}S_n,$ where $S_n=\sum_{k=1}^n\frac{1}{n}-\log(n+1),$ by linear forms in logarithms and harmonic numbers. In 1995, C. Elsner showed that slow…
The Euler-Mascheroni constant $\gamma=0.5772\dots\!$ is the $K=\mathbb{Q}$ example of an Euler-Kronecker constant $\gamma_K$ of a number field $K.$ In this note we consider the size of the $\gamma_q=\gamma_{K_q}$ for cyclotomic fields…
We present a new definition of Euler Gamma function. From the complex analysis and transalgebraic viewpoint, it is a natural characterization in the space of finite order meromorphic functions. We show how the classical theory and formulas…
The aim of this paper is to develop analytic techniques to deal with certain monotonicity of combinatorial sequences. (1) A criterion for the monotonicity of the function $\sqrt[x]{f(x)}$ is given, which is a continuous analog for one…
Convergent infinite products, indexed by all natural numbers, in which each factor is a rational function of the index, can always be evaluated in terms of finite products of gamma functions. This goes back to Euler. A purpose of this note…
We use a method, first developed for the Riemann zeta-function by Masser in ["Rational values of the Riemann zeta function", Journ. Num. Th. 131 (2011), 2037-2046], to prove a new zero estimate for polynomials in z and 1/Gamma(z). This…
Extending the Eulerian functions, we study their relationship with zeta function of several variables. In particular, starting with Weierstrass factorization theorem (and Newton-Girard identity) for the complex Gamma function, we are…
We provide representations of Euler's constant $\gamma=0.577...$ as series which converge geometrically fast (but use coefficients whose computation induces a quadratic cost). The asymptotic oscillations of these coefficients are discussed.
Let $H_k = 1 + 1/2 + 1/3 + \cdots + 1/k$ denote the $k$th harmonic number. We present an easy-to-implement algorithm for the computation of explicit closed-form evaluations, in terms of the digamma and polygamma functions, for Euler sums of…
We introduce a gamma function $\Ga(x,z)$ in two complex variables which extends the classical gamma function $\Ga(z)$ in the sense that $\lim_{x\to 1}\Ga(x,z)=\Ga(z)$. We will show that many properties which $\Ga(z)$ enjoys extend in a…
In the present paper we introduce some expansions, based on the falling factorials, for the Euler Gamma function and the Riemann Zeta function. In the proofs we use the Fa\'a di Bruno formula, Bell polynomials, potential polynomials,…
We have shown that in some region where the Euler integral of the first kind diverges, the Euler formula defines a generalized function. The connected of this generalized function with the Dirac delta function is found.
The Stieltjes constants $\gamma_k$ appear in the regular part of the Laurent expansion of the Riemman and Hurwitz zeta functions. We demonstrate that these coefficients may be written as certain summations over mathematical constants and…
Euler defines a function f(x) somehow as an infinite product and a generalization of [x], where [x] ist, what we now call following Legendre the Gamma-Funktion. He gets some recursive relationships for f(x), by applying some very nice…
This paper presents the evaluation of the Euler sums of generalized hyperharmonic numbers $H_{n}^{\left( p,q\right) }$ \[ \zeta_{H^{\left( p,q\right) }}\left( r\right) =\sum\limits_{n=1}^{\infty }\dfrac{H_{n}^{\left( p,q\right) }}{n^{r}}%…
Maximon has recently given an excellent summary of the properties of the Euler dilogarithm function and the frequently used generalizations of the dilogarithm, the most important among them being the polylogarithm function $Li_(z)$. The…
The Stieltjes constants $\gamma_k(a)$ appear as the coefficients in the regular part of the Laurent expansion of the Hurwitz zeta function $\zeta(s,a)$ about $s=1$. We present the evaluation of $\gamma_1(a)$ and $\gamma_2(a)$ at rational…