Related papers: Euler and the Gammafunction
In this paper, we introduce a way to generalize the Euler's gamma function as well as some related special functions. With a given polynomial in one variable $f(t)\ge 0$, we can associate a function, so-called "gamma function associated…
E661 in the Enestrom index. This was originally published as "Variae considerationes circa series hypergeometricas" (1776). In this paper Euler is looking at the asymptotic behavior of infinite products that are similar to the Gamma…
The attributes of Euler's constant Gamma have been a baffling problem to the world's mathematicians in the number theory field. In 1900, when German mathematician D. Hilbert addressed the 2nd International Congress of Mathematicians, he…
``In this paper we give the history of Leonhard Euler's work on the pentagonal number theorem, and his applications of the pentagonal number theorem to the divisor function, partition function and divergent series. We have attempted to give…
In the paper, we clarify some relations between solutions to the star-triangle equation via the gauge/YBE correspondence. We consider two solutions to the star-triangle relation in terms of Euler's gamma function. We derive these solutions…
Translation from the Latin original, "Inventio summae cuiusque seriei ex dato termino generali" (1735). E47 in the Enestrom index. In this paper Euler derives the Euler-Maclaurin summation formula, by expressing y(x-1) with the Taylor…
We generalize Romanoff's theorem. Also, we obtain a result on sums related to Euler's totient function.
A Thesis about Euler discussing the possibilities and limits of his method of work in Mathematics.
In this paper, we investigate the Euler-type integral representations for the generalized hypergeometric matrix function and develop some transformations in terms of hypergeometric matrix functions. Furthermore, unit and half arguments have…
In the first part we present results of four ``experimental'' determinations of the Euler-Mascheroni constant $\gamma$. Next we give new formulas expressing the $\gamma$ constant in terms of the Ramanujan-Soldner constant $\mu$. Employing…
The aim of the paper is to relate computational and arithmetic questions about Euler's constant $\gamma$ with properties of the values of the $q$-logarithm function, with natural choice of $q$. By these means, we generalize a classical…
In this paper we present a simple method for deriving an alternative form of the functional equation for Riemann's Zeta function. The connections between some functional equations obtained implicitly by Leonhard Euler in his work "Remarques…
This article surveys the Euler calculus - an integral calculus based on Euler characteristic - and its applications to data, sensing, networks, and imaging.
We introduce new generalizations of the Gamma and the Beta functions. Their properties are investigated and known results are obtained as particular cases.
We consider different generalizations of the Euler formula and discuss the properties of the associated trigonometric functions. The problem is analyzed from different points of view and it is shown that it can be formulated in a natural…
In this paper we will give a proof of a certain summation formula for Gamma functions utilizing Gegenbauer polynomials.
In this paper we discuss three types of the mean values of the Euler double zeta function. In order to get results we introduce three approximate formulas for this function.
Following an idea due to Euler, we evaluate the alternating sums of powers of consrcutive integers.
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 transformations of the sum identities for generalized harmonic and oscillatory numbers, obtained earlier in our recent report [1], enable us to derive the new identities expressed in terms of the corresponding square roots of x. At…