Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Meromorphic continuation of a q-analogue of multiple zeta function

Nita Tamang, Pitu Sarkar

TL;DR

This paper develops a comprehensive framework for the meromorphic continuation of a $q$-analogue of multiple zeta values. It introduces a translation formula that relates depth-$r$ zeta values to translates of depth $r-1$, proves normal convergence to justify analytic continuation, and then encodes the translation in a matrix form to explicitly locate poles and residues. The methodology yields a precise pole structure: all poles are simple and occur on hyperplanes defined by partial sums $s_1+\cdots+s_j$, with residues expressed in terms of lower-depth $q$-zeta values and bespoke coefficients $\mathcal L_n(t)$. The analysis is extended to the BZ-model, showing a parallel meromorphic continuation and pole description, and it touches variants such as SZ-star and dual models, highlighting the broad applicability of the translation-matrix approach.

Abstract

In this paper, we obtain the meromorphic continuation of a q-analogue of multiple zeta function using an elementary formula called translation formula. We then obtain the matrix representation of the translation formula and using it, we locate the poles of the function and the corresponding residues. While locating the poles, we also obtain an inverse of an infinite triangular matrix in a particular case.

Meromorphic continuation of a q-analogue of multiple zeta function

TL;DR

This paper develops a comprehensive framework for the meromorphic continuation of a -analogue of multiple zeta values. It introduces a translation formula that relates depth- zeta values to translates of depth , proves normal convergence to justify analytic continuation, and then encodes the translation in a matrix form to explicitly locate poles and residues. The methodology yields a precise pole structure: all poles are simple and occur on hyperplanes defined by partial sums , with residues expressed in terms of lower-depth -zeta values and bespoke coefficients . The analysis is extended to the BZ-model, showing a parallel meromorphic continuation and pole description, and it touches variants such as SZ-star and dual models, highlighting the broad applicability of the translation-matrix approach.

Abstract

In this paper, we obtain the meromorphic continuation of a q-analogue of multiple zeta function using an elementary formula called translation formula. We then obtain the matrix representation of the translation formula and using it, we locate the poles of the function and the corresponding residues. While locating the poles, we also obtain an inverse of an infinite triangular matrix in a particular case.
Paper Structure (5 sections, 13 theorems, 137 equations)

This paper contains 5 sections, 13 theorems, 137 equations.

Table of Contents

  1. Introduction
  2. Meromorphic continuation of $q$-multiple zeta function
  3. Matrix representation of the translation formula.
  4. Poles and Residues
  5. Meromorphic continuation of BZ model of $q$-multiple zeta function

Key Result

Proposition 1

Let $r$ be a positive integer and $(\alpha_1, \dots , \alpha_r)$ be an $r$-tuple of real numbers in $U_r$. Then the family of functions is normally convergent on the set $\mathcal{D}(\alpha_1, \dots , \alpha_r)=\{(s_1, \dots, s_r) \in \mathbb{C}^r: \text{Re}(s_j)\geq \alpha_j, 1 \leq j \leq r\}.$

Theorems & Definitions (27)

  • Proposition 1
  • proof
  • Corollary 1
  • proof
  • Proposition 2
  • proof
  • Theorem 1
  • Theorem 2
  • proof
  • Proposition 3
  • ...and 17 more