Related papers: On representations and differences of Stieltjes co…
We introduce a new technique for evaluation of series with zeta coefficients and also for evaluation of certain integrals involving the logGamma function. This technique is based on Hankel integral representations of the Hurwitz zeta, the…
We define a new class of generating function transformations related to polylogarithm functions, Dirichlet series, and Euler sums. These transformations are given by an infinite sum over the $j^{th}$ derivatives of a sequence generating…
In this article, we study the local behaviour of the multiple zeta functions at integer points and write down a Laurent type expansion of the multiple zeta functions around these points. Such an expansion involves a convergent power series…
In 2021, Hu and Kim defined a new type of gamma function $\widetilde{\Gamma}(x)$ from the alternating Hurwitz zeta function $\zeta_{E}(z,x)$, and obtained some of its properties. In this paper, we shall further investigate the function…
We establish the meromorphic continuation of certain multiple zeta functions of generalized Hurwitz type. From this meromorphic continuation, we obtain explicit formulas for their (derivative) values at nonpositive integers along a given…
For the multiple zeta function zeta2(s1,s2) of two variables,we obtain its integral representation(involving product of Hurwitz zeta functions) over the interval [1,infinity),with respect to second variable of Hurwitz zeta function and also…
A detailed study of a double integral representation of the Catalan's constant allows us to identify a duality identity for the Stieltjes transform on which it is based. This duality identity is then extended to an arbitrary dimensional…
The harmonic numbers and higher-order harmonic numbers appear frequently in several areas which are related to combinatorial identities, many expressions involving special functions in analytic number theory, and analysis of algorithms. The…
In our previous work we found sufficient conditions to be imposed on the parameters of the generalized hypergeometric function in order that it be completely monotonic or of Stieltjes class. In this paper we collect a number of consequences…
Integrals involving the kernel function $sech (\pi x)$ over a semi-infinite range are of general interest in the study of Riemann's function $\zeta(s)$ and Hurwitz' function $\zeta(s,a)$. Such integrals that include the $arctan$ and $log$…
A converse method to the Construction of Salem (1945) of convergent families of Salem numbers is investigated in terms of an association between Salem polynomials and Hurwitz quotients via expansive polynomials of small Mahler measure. This…
In this article, we develop a k-free zeta Dirichlet series into a Laurent series with a simple pole, and prove a Stieltjes like formula for the expansion coefficients of the regular part. We also investigate another analytical continuation…
We present a large number of analytic evaluations of Euler sums, namely sums such as \begin{align} M(m,n_0,n_1,n_2, \ldots, n_t) &= \sum_{k=1}^\infty \frac{H(k)^m}{k^{n_0} (k+1)^{n_1} (k+2)^{n_2} \cdots (k+t)^{n_t}}, \nonumber \end{align}…
We study analytic and geometric properties of Stieltjes and inverse Stieltjes families defined on a separable Hilbert space and establish various minimal representations for them by means of compressed resolvents of various types of linear…
We obtain a new proof of Hurwitz's formula for the Hurwitz zeta function $\zeta(s, a)$ beginning with Hermite's formula. The aim is to reveal a nice connection between $\zeta(s, a)$ and a special case of the Lommel function $S_{\mu,…
The Riemann zeta identity at even integers of Lettington, along with his other Bernoulli and zeta relations, are generalized. Other corresponding recurrences and determinant relations are illustrated. Another consequence is the application…
We review generalized zeta functions built over the Riemann zeros (in short: "superzeta" functions). They are symmetric functions of the zeros that display a wealth of explicit properties, fully matching the much more elementary Hurwitz…
The Hlawka Zeta Function is a Dirichlet series defined geometrically which provides an integral representation of the number of lattice points contained in the dilation $tD$ for some star shaped region $D\subset \mathbb{R}^{2}$ and some…
For Hurwitz Zeta function,we consider its Taylor series expansion about various points as an analytic function of second variable in appropriate discs.We show that these Taylor are all polynomials in second variable for a non positive…
We obtain a characterization of generalized Stieltjes functions of any order \lambda > 0 in terms of inequalities for their derivatives on (0,\infty). When \lambda=1, this provides a new and simple proof of a characterization of Stieltjes…