Related papers: Identities for number series and their reciprocals…
By means of partial fraction method, we investigate the decomposition of rational functions. Several striking identities on harmonic numbers and generalized Apery numbers will be established, including the binomial-harmonic number identity…
When the discriminants $\Delta$ and $\Delta p^2$ are idoneal, Patane proved a theorem which connects the theta series associated to binary quadratic forms of each discriminant. This paper generalizes the main theorem of Patane by no longer…
We develop a systematic and fully explicit approach to the evaluation of binomial sums involving reciprocals of binomial coefficients based on Beta integral techniques. Starting from a simple integral representation, we provide a derivation…
Fix $k \ge 3$. If a multiplicative function $f$ satisfies \[ f(x_1+x_2+\dots+x_k) = f(x_1) + f(x_2) + \dots + f(x_k) \] for arbitrary positive triangular numbers $x_1, x_2, \dots, x_k$, then $f$ is the identity function. This extends Chung…
In this series of seven papers, predominantly by means of elementary analysis, we establish a number of identities related to the Riemann zeta function. Whilst this paper is mainly expository, some of the formulae reported in it are…
We establish some functional identities of theta functions, an elementary proof of classical fourth-order identities, Landen transformations, and q series from the eigenvectors of the discrete Fourier transform. Also, we derive connection…
This is a short review of some recent results obtained by the author. These results are related the problem of obtaining polynomial identities (computational formulas) for some matrix functions by means of the known polarization theorem,…
In this study, we establish a significant connection between certain subclasses of complex order univalent functions and the Mittag-Leffler-type Poisson distribution series. We provide criteria for these series to belong to the specific…
We prove a recursive identity involving formal iterated logarithms and formal iterated exponentials. These iterated logarithms and exponentials appear in a natural extension of the logarithmic formal calculus used in the study of…
The prime-counting function $\pi(x)$ which returns the number of primes smaller or equal to a given number is a topic of interest in number theory. An algorithm based on a cyclic group isomorphic to $Z/nZ$, the so-called $Z$-functions, was…
In this paper, the asymptotic distributions of estimators for the regularized functional canonical correlation and variates of the population are derived. The method is based on the possibility of expressing these regularized quantities as…
An important class of fractional differential and integral operators is given by the theory of fractional calculus with respect to functions, sometimes called $\Psi$-fractional calculus. The operational calculus approach has proved useful…
The main goal of this research is to model and investigate generalizations of functions from [31]. Arguments of modeled functions are presented by the representation $\pi_{\mathfrak p}$ from [22].
It is the purpose of this article to outline a course that can be given to engineers looking for an understandable mathematical description of the foundations of distribution theory and the necessary functional analytic methods. Arguably,…
In this paper, we describe the line Dirac delta function of a curve in three-dimensional space in terms of the distance function to the curve. Its extension to level set formulation and plane curves are also developed. The main ideas can be…
We derive an identity for certain linear combinations of polylogarithm functions with negative exponents, which implies relations for linear combinations of Eulerian numbers. The coefficients of our linear combinations are related to…
We utilize a combination of integral transforms, including the Laplace transform, with some classical results in analytic number theory concerning the Riemann $\xi$-function, to obtain a new integral equation. We also provide a new proof of…
In fractional calculus there are two approaches to obtain fractional derivatives. The first approach is by iterating the integral and then defining a fractional order by using Cauchy formula to obtain Riemann fractional integrals and…
The overlap Dirac operator at nonzero quark chemical potential involves the computation of the sign function of a non-Hermitian matrix. In this talk we present an iterative method, first proposed by us in Ref. [1], which allows for an…
We establish some identities in law for the convolution of a beta prime distribution with itself, involving the square root of beta distributions. The proof of these identities relies on transformations on generalized hypergeometric series…