Related papers: Euler and the Multiplication Formula for the Gamma…
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…
An interplay between the Lambert series and Euler's Pentagonal Number Theorem gives an Euler-type recurrence relation for any given arithmetical function. As consequences of this, we present Euler-type recurrence relations for some…
In this paper new $\Gamma$-functional is constructed upon the basis of the set of almost linear increments of the Hardy-Littlewood integral. This functional generates a $\Gamma$-equivalent of the Fermat-Wiles theorem and also new set of…
In this paper, we prove Newton-Maclaurin type inequalities for functions obtained by linear combination of two neighboring primary symmetry functions, which is a generalization of the classical Newton-Maclaurin inequality.
The Dirichlet lambda function $\lambda(s)$ is defined for $\mathrm{Re}(s) > 1$ by \[ \lambda(s) = \sum_{n=0}^{\infty} \frac{1}{(2n+1)^s}. \] This function was initially studied by Euler on the real line, where he denoted it by $N(s)$. In…
We generalize our previous new definition of Euler Gamma function to higher Gamma functions. With this unified approach, we characterize Barnes higher Gamma functions, Mellin Gamma functions, Barnes multiple Gamma functions, Jackson…
The generalized Euler number E_{n|k} counts the number of permutations of {1,2,...,n} which have a descent in position m if and only if m is divisible by k. The classical Euler numbers are the special case when k=2. In this paper, we study…
We transformed the generalized exponential power series to another functional form suitable for further analysis. By applying the Cauchy-Euler differential operator in the form of an exponential operator, the series became a sum of…
We give another proof for \[ \sum_{n=1}^{\infty}\frac{1}{n^2}=\frac{\pi^2}{6} \] that basically follows from the theory of difference equations.
A GGC (Generalized Gamma Convolution) representation of Riemann's Xi-function is constructed.
In 1922, Harald Bohr and Johannes Mollerup established a remarkable characterization of the Euler gamma function using its log-convexity property. A decade later, Emil Artin investigated this result and used it to derive the basic…
In this paper, we obtain some formulas for double nonlinear Euler sums involving harmonic numbers and alternating harmonic numbers. By using these formulas, we give new closed form sums of several quadratic Euler series through Riemann zeta…
The recurrence matrix relations, differentiation formulas, and analytical and fractional integral properties of incomplete gamma matrix functions $\gamma(Q, x)$ and $\Gamma(Q, x)$ are all covered in this article. The generalized incomplete…
We derive a closed form solution for the Kullback-Leibler divergence between two generalized gamma distributions. These notes are meant as a reference and provide a guided tour towards a result of practical interest that is rarely…
The method of exhaustion is generalized to a simple formula that can be used to integrate functions under very general conditions, provided that the integral exists. Both a geometric proof (following the usual procedure for the method of…
The sum formula is a basic identity of multiple zeta values that expresses a Riemann zeta value as a homogeneous sum of multiple zeta values of a given dimension. This formula was already known to Euler in the dimension two case,…
In this paper we give some interesting identities between Euler numbers and zeta functions. Finally we will give the new values of Euler zeta function at positive even integers.
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…
Based on elementary methods and techniques, the explicit formula for the generalized Euler function $\varphi_{e}(n)(e=8,12)$ is given, and then a sufficient and necessary condition for $\varphi_{8}(n)$ or $\varphi_{12}(n)$ to be odd is…
Euler discovered recurrence for divisor sum functions as a consequence of the pentagonal numbers theorem. With similar idea and also motivated by Ewell's work in 1977, we prove new recurrences for certain divisor sum functions and…