Related papers: Euler and the Gammafunction
In this paper, we mainly show that Euler sums of generalized hyperharmonic numbers can be expressed in terms of linear combinations of the classical Euler sums.
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.
The aim of this note is to gather formal similarities between two apparently different functions; {\em Euler's function} $\Gamma$ and {\em Anderson-Thakur function} $\omega$. We discuss these similarities in the framework of the {\em…
In this paper we extend the notion of the Euler characteristic to persistent homology and give the relationship between the Euler integral of a function and the Euler characteristic of the function's persistent homology. We then proceed to…
The aim of this paper is to derive on the basis of the Euler's formula several analytical relations which hold for certain classes of planar graphs and which can be useful in algorithmic graph theory.
An inequality concerning ratios of gamma functions is proved. This answers a question of Guo and Qi (2003).
In this paper, we introduce the polynomial continued fraction, a close relative of the well-known simple continued fraction expansions which are widely used in number theory and in general. While they may not possess all the intriguing…
Translation of "Methodus succincta summas serierum infinitarum per formulas differentiales investigandi" (1780). Euler wants to represent some given series of functions S(x)=X(x)+X(x+1)+X(x+2)+etc. in a different way. He writes S as a…
We introduce and prove several new formulas for the Euler-Mascheroni Constant. This is done through the introduction of the defined E-Harmonic function, whose properties, in this paper, lead to two novel formulas, alongside a family of…
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…
The elliptic gamma function is a generalization of the Euler gamma function. Its trigonometric and rational degenerations are the Jackson q-gamma function and the Euler gamma function. We prove multiplication formulas for the elliptic gamma…
In the present article, we study Bell based Euler polynomial of order {\alpha} and investigate some useful correlation formula, summation formula and derivative formula. Also, we introduce some relation of string number of the second kind.…
Recently, T. Kim considered Euler zeta function which interpolates Euler polynomials at negative integer (see [3]). In this paper, we study degenerate Euler zeta function which is holomorphic function on complex s-plane associated with…
We analyze the issue of the interpretation of the wavefunction, namely whether it should be interpreted as describing individual systems or ensembles of identically prepared systems. We propose an experiment which can decide the issue,…
In this paper, we give a short elementary proof of the well known Euler's recurrence formula for the Riemann zeta function at positive even integers and integral representations of the Riemann zeta function at positive integers and at…
In this essay I will give a strictly subjective selection of different types of zeta functions. Instead of providing a complete list, I will rather try to give the central concepts and ideas underlying the theory. This article is going to…
Let $T$ be the triangle with vertices (1,0), (0,1), (1,1). We study certain integrals over $T$, one of which was computed by Euler. We give expressions for them both as a linear combination of multiple zeta values, and as a polynomial in…
In this paper, we propose a numerical method of Fourier transform based on hyperfunction theory. In the proposed method, we compute analytic functions called the defining functions, which give the desired Fourier transform as a…
Generalisations of the classical Euler formula to the setting of fractional calculus are discussed. Compound interest and fractional compound interest serve as motivation. Connections to fractional master equations are highlighted. An…
In this paper, we study the holomorphic function defined by the infinite product $\Gamma_{a,r}(s) =\prod_{n \geq 0} (1 + \frac{1}{a+ nr})^s (1 + \frac{s}{a+nr})^{-1}$ which generalize Euler's definition in the sense that $\Gamma(s) =…