A combinatorial identity on Galton-Watson process

TL;DR

This paper proves a key combinatorial identity related to Galton-Watson processes using elementary methods, extending its validity to all positive real parameters and highlighting its relevance in random graph theory.

Contribution

It provides two elementary proofs of a fundamental identity in Galton-Watson processes, broadening its applicability to all positive real numbers.

Findings

Established the identity for all positive real m and c.

Provided combinatorial and power-serial proofs.

Linked the identity to random graph theory applications.

Abstract

Let $f(m,c)=\sum_{k=0}^{\infty} (km+1)^{k-1} c^k e^{-c(km+1)/m} / (m^kk!)$. For any positive integer $m$ and positive real $c$, the identity $f(m,c)=f(1,c)^{1/m}$ arises in the random graph theory. In this paper, we present two elementary proofs of this identity: a pure combinatorial proof and a power-serial proof. We also proved that this identity holds for any positive reals $m$ and $c$.