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

Dylan Müller

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.

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

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 for a natural completion of the zeta function.
Paper Structure (11 sections, 10 theorems, 85 equations, 2 figures, 1 table)

This paper contains 11 sections, 10 theorems, 85 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. The heat kernel and the spectral zeta function on $T_{q+1}$
  3. The fundamental symmetry and explicit formulas for $G_\pm$
  4. Proof of \ref{['Thm:Fundamental symmetry']}
  5. Kesten's generating function and a formula for $G_-$
  6. The formula for $G_+$
  7. Proof of \ref{['Thm:Main']} and a recursive relation for the $P_n$'s
  8. Proof of \ref{['Thm:Main']}
  9. A recursive relation for the $P_n$'s
  10. The functional equation
  11. Combinatorial interpretation of $P_n$

Key Result

Theorem 1.1

For all integers $n \ge 1$ and $q \ge 2$, we have where $P_n$ is a palindromic and monic polynomial of degree $2n-2$ with integer coefficients. Moreover, the coefficients of $P_n$ count weighted $2$-coloured Dyck words, and hence are non-negative.

Figures (2)

  • Figure 1: Example of a Dyck path of length $8$.
  • Figure 2: Example of a $2$-coloured Dyck path of length $12$.

Theorems & Definitions (18)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Proposition 1.4: K25
  • Remark 3.1
  • Lemma 3.2
  • proof
  • Remark 4.1
  • Corollary 4.2: of \ref{['Lem:Elementary expression for L']}
  • Proposition 5.1
  • ...and 8 more