Related papers: Tau Function Approach to Theta Functions
Starting from equations obeyed by functions involving the first or the second derivatives of the biconfluent Heun function, we construct two expansions of the solutions of the biconfluent Heun equation in terms of incomplete Beta functions.…
Observing a multiple version of the divisor function we introduce a new zeta function which we call a multiple finite Riemann zeta function. We utilize some $q$-series identity for proving the zeta function has an Euler product and then,…
As a generalization of the Dedekind zeta function, Weng defined the high rank zeta functions and proved that they have standard properties of zeta functions, namely, meromorphic continuation, functional equation, and having only two simple…
A basic feature of Teichm\"uller theory of Riemann surfaces is the interplay of two dimensional hyperbolic geometry, the behavior of geodesic-length functions and Weil-Petersson geometry. Let $\mathcal{T}_g$ $(g\geq 2)$ be the Teichm\"uller…
A Master equation has been previously obtained which allows the analytic integration of a fairly large family of functions provided that they possess simple properties. Here, the properties of this Master equation are explored, by extending…
Properties of four quintic theta functions are developed in parallel with those of the classical Jacobi null theta functions. The quintic theta functions are shown to satisfy analogues of Jacobi's quartic theta function identity and…
We consider a variant expression to regularize the Euler product representation of the zeta functions, where we mainly apply to that of the Riemann zeta function in this paper. The regularization itself is identical to that of the zeta…
The generalization of new mock theta functions of Andrews and Bringmann et al are given. Further we have given the expansion of these bilateral generalized new mock theta functions as 2 phi 1 series by Slaters transformation. After that we…
In arXiv:2211.09069, significant progress was made in decomposing simple products of fermion correlation functions, and in summing over spin structures of superstring amplitudes in genus two under cyclic constraints. In this manuscript we…
In this paper, we introduce and study two new types of non-abelian zeta functions for curves over finite fields, which are defined by using (moduli spaces of) semi-stable vector bundles and non-stable bundles. A Riemann-Weil type hypothesis…
We consider the asymptotic expansion of the functional series \[S_{\mu,\gamma}(a;\lambda)=\sum_{n=1}^\infty \frac{n^\gamma e^{-\lambda n^2/a^2}}{(n^2+a^2)^\mu}\] for real values of the parameters $\gamma$, $\lambda>0$ and $\mu\geq0$ as…
A theta surface in affine 3-space is the zero set of a Riemann theta function in genus 3. This includes surfaces arising from special plane quartics that are singular or reducible. Lie and Poincar\'e showed that theta surfaces are precisely…
In this article, we explore a natural extension of the quadratic parametrization introduced in our previous work. By replacing the integer $n$ by $n^s$ ($ s\in\mathbb{R}, s>1$) and allowing the parameters to be real, we obtain for each…
We study Riemann-type functional equations with respect to value-distribution theory and derive implications for their solutions. In particular, for a fixed complex number $a\neq0$ and a function from the Selberg class $\mathcal{L}$, we…
Taylor expansions of analytic functions are considered with respect to two points. Cauchy-type formulas are given for coefficients and remainders in the expansions, and the regions of convergence are indicated. It is explained how these…
Let $X_1,\dots, X_n$ be i.i.d. random variables sampled from a normal distribution $N(\mu,\Sigma)$ in ${\mathbb R}^d$ with unknown parameter $\theta=(\mu,\Sigma)\in \Theta:={\mathbb R}^d\times {\mathcal C}_+^d,$ where ${\mathcal C}_+^d$ is…
We examine "partition zeta functions" analogous to the Riemann zeta function but summed over subsets of integer partitions. We prove an explicit formula for a family of partition zeta functions already shown to have nice properties -- those…
The main purpose of this paper is to show some relations between the Riemann zeta function and the generalized Bernoulli polynomials of level $m$. Our approach is based on the use of Fourier expansions for the periodic generalized Bernoulli…
In this paper we establish the relationships between theta functions of arbitrary order and their derivatives. We generalize our previous work math.AG/0310085 and prove that for any n>1 the map sending an abelian variety to the set of Gauss…
In arXiv:hep-th/0310113 we started a program of creating a reference-book on matrix-model tau-functions, the new generation of special functions, which are going to play an important role in string theory calculations. The main focus of…