A note on Puder's generalised co-growth formula for trees

TL;DR

This paper proves a conjecture extending Puder's co-growth formula to bi-regular trees, introduces a resolvent identity as an operator analogue, and simplifies the proof for regular trees.

Contribution

It extends Puder's co-growth formula to bi-regular trees and introduces a resolvent identity, providing a new operator perspective and simplifying existing proofs.

Findings

Proved the conjecture for bi-regular trees

Established a resolvent identity as an operator version

Provided a simpler proof for regular trees

Abstract

In this note, we prove a conjecture of Puder on an extension of the co-growth formula to any non-negative function defined on a bi-regular tree. A key component of our proof is the establishment of a resolvent identity, which serves as an operator version of the co-growth formula. We also provide a simpler proof of Puder's generalised co-growth formula for the regular tree.