Multiple zeta functions at regular integer points

TL;DR

This paper develops recurrence relations for multiple zeta functions at integer points, enabling explicit calculation of their values as rational linear combinations of multiple zeta values, extending previous results.

Contribution

It introduces new recurrence relations for multiple zeta functions at integer points, generalizing earlier work and providing a method for explicit value computation.

Findings

Recurrence relations for multiple zeta functions at integer points.

Explicit method to compute special values as rational sums.

Extension of previous results by Akiyama-Egami-Tanigawa and Matsumoto.

Abstract

We show the recurrence relations of the Euler-Zagier multiple zeta-function which describes the $r$-fold function with one variable specialized to a non-positive integer as a rational linear combination of $(r-1)$-fold functions, which extends the previous results of Akiyama-Egami-Tanigawa and Matsumoto. As an application, we obtain an explicit method to calculate the special values of the multiple zeta-function at any integer point (the arguments could be neither all-positive nor all-non-positive) as a rational linear summation of the multiple zeta values.