This paper investigates a generalization the classical Hurwitz zeta function. It is shown that many of the properties exhibited by this special function extends to class of functions called hypergeometric Hurwitz zeta functions, including their analytic continuation to the complex plane and a pre-functional equation satisfied by them. As an application, a formula for moments of hypergeometric H...

Motivated by results for the HCIZ integral in Part I of this paper, we study the structure of monotone Hurwitz numbers, which are a desymmetrized version of classical Hurwitz numbers. We prove a number of results for monotone Hurwitz numbers and their generating series that are striking analogues of known results for the classical Hurwtiz numbers. These include explicit formulas for monotone Hu...

Double Hurwitz numbers count branched covers of CP with fixed branch points, with simple branching required over all but two points 0 and∞, and the branching over 0 and∞ points specified by partitions of the degree (withm and n parts respectively). Single Hurwitz numbers (or more usually, Hurwitz numbers) have a rich structure, explored by many authors in fields as diverse as algebraic geometry...

Hurwitz numbers count genus g, degree d covers of P1 with fixed branch locus. This equals the degree of a natural branch map defined on the Hurwitz space. In tropical geometry, algebraic curves are replaced by certain piece-wise linear objects called tropical curves. This paper develops a tropical counterpart of the branch map and shows that its degree recovers classical Hurwitz numbers. Furthe...

Abstract: Motivated largely by a number of recent investigations, we introduce and investigate the various properties of a certain new family of the λ -generalized Hurwitz-Lerch zeta functions. We derive many potentially useful results involving these λ -generalized Hurwitz-Lerch zeta functions including (for example) their partial differential equations, new series and Mellin-Barnes type conto...

We derive a closed-form expression for all genus 1 Hurwitz numbers, and give a simple new graph-theoretic interpretation of Hurwitz numbers in genus 0 and 1. (Hurwitz numbers essentially count irreducible genus g covers of the sphere, with arbitrary specified branching over one point, simple branching over other specified points, and no other branching. The problem is equivalent to counting tra...

We derive a closed-form expression for all genus 1 Hurwitz numbers, and give a simple new graph-theoretic interpretation of Hurwitz numbers in genus 0 and 1. (Hurwitz numbers essentially count irreducible genus g covers of the sphere, with arbitrary specified branching over one point, simple branching over other specified points, and no other branching. The problem is equivalent to counting tra...

We derive a closed-form expression for all genus 1 Hurwitz numbers, and give a simple new graph-theoretic interpretation of Hurwitz numbers in genus 0 and 1. (Hurwitz numbers essentially count irreducible genus g covers of the sphere, with arbitrary specified branching over one point, simple branching over other specified points, and no other branching. The problem is equivalent to counting tra...

Hurwitz loci have played a basic role in the study of themoduli space of curves at least since 1872 when Clebsch proved that Mg is irreducible by showing that a certain Hurwitz space is connected. Hurwitz cycles on Mg are essential in the work of Harris and Mumford [HM] on the Kodaira dimension of Mg and are expected to govern the length of minimal affine stratifications of Mg. Faber and Pandha...

In this study, we obtain new classes of linear codes over Hurwitz integers equipped with a new metric. We refer to the metric as Hurwitz metric. The codes with respect to Hurwitz metric use in coded modulation schemes based on quadrature amplitude modulation (QAM)-type constellations, for which neither Hamming metric nor Lee metric. Also, we define decoding algorithms for these codes when up to...