# Analytic Continuation for Multiple Zeta Values using Symbolic Representations

## TL;DR

This paper develops a symbolic method to analytically continue multiple zeta values at negative indices, extending known results and connecting to Bernoulli polynomials and Faulhaber's formula.

## Contribution

It introduces a novel symbolic representation for harmonic sums at negative indices, enabling new recurrence relations, generating functions, and reinterpretations of analytic continuation.

## Key findings

- Recovered and extended recurrence relations for harmonic sums
- Connected analytic continuation to Bernoulli polynomials and Faulhaber's formula
- Provided a natural renormalization perspective for multiple zeta functions

## Abstract

We introduce a symbolic representation of $r$-fold harmonic sums at negative indices. This representation allows us to recover and extend some recent results by Duchamp et al., such as recurrence relations and generating functions for these sums. This approach is also applied to the study of the family of extended Bernoulli polynomials, which appear in the computation of harmonic sums at negative indices. It also allows us to reinterpret the Raabe analytic continuation of the multiple zeta function as both a constant term extension of Faulhaber's formula, and as the result of a natural renormalization procedure for Faulhaber's formula.