Related papers: Integral Representations for Multiple Ap\'ery-Like…
We describe the solutions to a family of rotationally symmetric second order partial differential equations in the complex plane that arises from a four-dimensional complex Lie algebra whose spanning set generates the algebra from which…
In the present article, the author uses Fourier theory of tempered distributions (generalized functions) in deriving a formula for Dirichlet-like integrals. The applied method is remarkably efficient and allows a solution in a few…
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…
In this paper, we employ the theories and techniques of hypergeometric functions to provide two distinct proofs of the conjectured identities involving multiple Ap\'ery-like series with central binomial coefficients and multiple harmonic…
In this paper, we find new integral representations for the generalized Hermite linear functional in the real line and the complex plane. As an application, new integral representations for the Euler Gamma function are given.
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…
In the paper, the author elementarily unifies and generalizes eight identities involving the functions $\frac{\pm1}{e^{\pm t}-1}$ and their derivatives. By one of these identities, the author establishes two explicit formulae for computing…
A new family of polynomials, called cumulant polynomial sequence, and its extensions to the multivariate case is introduced relied on a purely symbolic combinatorial method. The coefficients of these polynomials are cumulants, but depending…
In this paper, we derive a general formula to express the product of three theta functions as a linear combination of other products of three theta functions. Moreover, we use the main formula to deduce a general formula for the product of…
We give a representation of the classical theory of multiplicative arithmetic functions (MF)in the ring of symmetric polynomials. The basis of the ring of symmetric polynomials that we use is the isobaric basis, a basis especially sensitive…
We prove an identity between three infinite families of polynomials which are defined in terms of `bosonic', `fermionic', and `one-dimensional configuration' sums. In the limit where the polynomials become infinite series, they give…
We find that the solution of the polar angular differential equation can be written as the universal associated Legendre polynomials. Its generating function is applied to obtain an analytical result for a class of interesting integrals…
Euler's sum formula and its multi-variable and weighted generalizations form a large class of the identities of multiple zeta values. In this paper we prove a family of identities involving Bernoulli numbers and apply them to obtain…
We describe a new infinite family of multi-parameter functional equations for the Rogers dilogarithm, generalizing Abel's and Euler's formulas. They are suggested by the Thermodynamic Bethe Ansatz approach to the Renormalization Group flow…
Using a polylogarithmic identity, we express the values of $\zeta$ at odd integers $2n+1$ as integrals over unit $n-$dimensional hypercubes of simple functions involving products of logarithms. We also prove a useful property of those…
We introduce the central Fubini-like numbers and polynomials using Rota approach. Several identities and properties are established as generating functions, recurrences, explicit formulas, parity, asymptotics and determinantal…
Matrix Lie groups provide a language for describing motion in such fields as robotics, computer vision, and graphics. When using these tools, we are often faced with turning infinite-series expressions into more compact finite series (e.g.,…
In this paper, we introduce a class of Dirichlet series defined in terms of the Riemann zeta-function, motivated by the study of their special values, and establish integral representations for these series. We also define an extension of…
We create a sequence version of calculus. First, we define equivalence, some fundamental operations, differential, and integral for sequences. Then, we propose sequence versions of identity function, power function, exponential function,…
We compute explicit formulae for Dirichlet generating functions enumerating finite-dimensional irreducible complex representations of potent and saturable principal congruence subgroups of $\mathrm{SL}_4^m(\mathfrak{o})$ ($m\in\mathbb{N}$)…