The zeta function of regular trees, their special values and functional equations

TL;DR

This paper computes special values of the spectral zeta function on regular trees, revealing explicit formulas, symmetries, and a functional equation similar to classical zeta functions.

Contribution

It provides explicit formulas for the zeta function's special values, uncovers symmetries, and establishes a functional equation for the spectral zeta function on regular trees.

Findings

Explicit formulas for special zeta values in terms of palindromic polynomials

Discovery of symmetries between positive and negative integer values

Establishment of a functional equation for the zeta function

Abstract

We determine the special values at positive integers of the spectral zeta function associated with the combinatorial Laplacian on the regular tree. These values admit explicit formulas in terms of certain polynomials, which we show to be palindromic and to have non-negative integer coefficients with a combinatorial interpretation. Along the way, we uncover unexpected symmetries between the values of the zeta function at negative and positive integers, expressed at the level of their generating functions. Using these symmetries, we ultimately establish a functional equation of the type \( s \longleftrightarrow 1-s \) for a natural completion of the zeta function.