Related papers: A note on a generalized circular summation formula…
By some hypergeometric summation theorems, the authors establish a series of new infinite summation formulas involving generalized harmonic numbers related to Riemann-Zeta function, with three different patterns.
We exploit transformations relating generalized $q$-series, infinite products, sums over integer partitions, and continued fractions, to find partition-theoretic formulas to compute the values of constants such as $\pi$, and to connect sums…
Extending Sellers' result, Das et al. recently proved some congruence results for generalized overcubic partitions using theta functions and posed some related conjectures. In this paper, we provide a combinatorial proof of a result in…
In this article we present certain formulas involving arithmetical functions. In the first part we study properties of sums and product formulas for general type of arithmetic functions. In the second part we apply these formulas to the…
Partial zeta functions of algebraic varieties over finite fields generalize the classical zeta function by allowing each variable to be defined over a possibly different extension field of a fixed finite field. Due to this extra variation…
We show how the generalized Lambert series sum(n>=1, x*q^n/(1-x*q^n)) can be computed with Theta convergence. This allows the computation of the sum of the inverse Fibonacci numbers without splitting the sum into even and odd part. The…
The sum formula for $q$-multiple zeta values is a well-known relation. In this paper, we present its generalization for the $q$-multiple zeta function.
This is a discussion of miscellaneous summation, integration and transformation formulas obtained using Fourier analysis. The topics covered are: Series of the form $\sum_{n\in\mathbb{Z}} c_ne^{\pi i \gamma n^2}$; Fusion of integrals, and…
We introduce a one parameter deformation of Zwegers' multivariable $\mu$-function by applying iterations of the $q$-Borel summation method, which is also a multivariate analogue of the generalized $\mu$-function introduced by the authors.…
In this brief note, we show how to apply Kummer's and other quadratic transformation formulas for Gauss' and generalized hypergeometric functions in order to obtain transformation and summation formulas for series with harmonic numbers that…
Contour integral representations for Riemann's Zeta function and Dirichelet's Eta (alternating Zeta) function are presented and investigated. These representations flow naturally from methods developed in the 1800's, but somehow they do not…
Ohno-Wakabayashi's cyclic sum formula for multiple zeta-star values is generalized by Igarashi with one or two parameters. In this article, we give a possible answer for one of his problems about a generalization with three parameters.
We obtain formulas for the coefficients of positive and negative powers of a partial theta function.
This paper develops a generalized cotangent-type series, extending classical expansions to higher-order lattice sums. By introducing a new family of series indexed by integer powers, we derive closed form representations that combine…
A generalization of the classical Lipschitz summation formula is proposed. It involves new polylogarithmic rational functions constructed via the Fourier expansion of certain sequences of Bernoulli--type polynomials. Related families of…
We generalize the standard Poisson summation formula for lattices so that it operates on the level of theta series, allowing us to introduce noninteger dimension parameters (using the dimensionally continued Fourier transform). When…
We prove a new linear relation for multiple zeta values. This is a natural generalization of the restricted sum formula proved by Eie, Liaw and Ong. We also present an analogous result for finite multiple zeta values.
Generalized product formulas and index transforms, involving products of Whittaker's functions of different indices are established and investigated. The corresponding inversion formulas are found. Particular cases cover index transforms…
We shall show that the sum of the series formed by the so-called hyperharmonic numbers can be expressed in terms of the Riemann zeta function. More exactly, we give summation formula for the general hyperharmonic series.
The sum formula is a well known relation in the field of the multiple zeta values. In this paper, we present its generalization for the Euler-Zagier multiple zeta function.