Related papers: Zeta-Functions and Star-Products
In this paper, we discuss a similar functional to that of a standard integral. The main difference is in its definition: instead of taking a sum, we are taking a product. It turns out this new "star-integral" may be written in terms of the…
We evaluate the multiple zeta values $\zeta(\{2\}^k)$ by proving a certain factorization property. The proof uses a combinatorial bijection and elementary telescoping series. We show how the infinite product for the sine function in fact…
On the one hand the Fermi-Dirac and Bose-Einstein functions have been extended in such a way that they are closely related to the Riemann and other zeta functions. On the other hand the Fourier transform representation of the gamma and…
The purpose of this paper is to investigate coefficient matrices of functional equations of zeta functions associated with homogeneous cones, which are given explicitly in the previous paper, in detail. We prove that the coefficient matrix…
We present a pedagogical exposition of some applications of functional methods in quantum field theory: we use heat-kernel and zeta-function techniques to study the Casimir effect, the pair production in strong electric fields, quantum…
This paper explores the interplay between holonomy, Ihara zeta functions, and cohomological structures within the framework of ratified F-completions of foliated manifolds. We develop a novel formalism for the Gamma-set, a topological…
This work prolongs, using an operator method, the investigations started in our recent paper J. Math. Phys. 51., 102108 on the spectrum and states of the harmonic oscillator on twisted Moyal plane, where rather a Moyal-star-algebraic…
We obtain recursive formulas for the stuffle product of multiple zeta values and of multiple zeta-star values. Then we apply the formulas to prove several stuffle product formulas with one or two strings of $z_p$'s. We also describe how to…
Properties of the functional classes of star-product elements associated with higher-spin gauge fields and gauge parameters are elaborated. Cohomological interpretation of the nonlinear higher-spin equations is given. An algebra ${\mathcal…
In this paper we obtain new properties of a signal generated by the Riemann zeta-function on the critical line. At the same time we obtain an asymptotic formula for a new class of transcendental integrals connected with the Riemann…
In this paper we study the zeta functions associated to the minimal spherical principal series of representations for a class of reductive p-adic symmetric spaces, which are realized as open orbits of some prehomogeneous spaces. These…
Siegel defined zeta functions associated with indefinite quadratic forms, and proved their analytic properties such as analytic continuations and functional equations. Coefficients of these zeta functions are called measures of…
It is known that the values of multiple zeta functions (MZFs) at non-positive integers can be expressed by Bernoulli numbers. This paper gives explicit formulas for the values of MZFs and multiple zeta star functions (MZSFs) at non-positive…
We identify a partition-theoretic generalization of Riemann zeta function and the equally positive integer-indexed harmonic sums at infinity, to obtain the generating function and the integral representations of the latter. The special…
While the definition of a fractional integral may be codified by Riemann and Liouville, an agreed-upon fractional derivative has eluded discovery for many years. This is likely a result of integral definitions including numerous constants…
We define the generalized Dirichlet beta and Riemann zeta functions in terms of the integrals, involving powers of the hyperbolic secant and cosecant functions. The corresponding functional equations are established. Some consequences of…
We introduce two kinds of fractional integral operators; the one is defined via the exponential-integral function $$ E_1(x)=\int_x^\infty \frac{e^{-t}}{t}\,dt,\quad x>0, $$ and the other is defined via the special function $$…
In this paper,we develop a novel representation of the zeta function expressed as the limiting difference between two structured double sums. This approach leads to a new and elegant identity involving maximum functions and additive terms,…
Euler discovered a formula for expressing the value of the Riemann zeta function for all even positive integer arguments. A closed-form expression for the Riemann zeta function for all odd integer arguments, based on the values of the…
This paper investigates a new family of special functions referred to as hypergeometric zeta functions. Derived from the integral representation of the classical Riemann zeta function, hypergeometric zeta functions exhibit many properties…