A combinatorial identity on Galton-Watson process

2015/06/28 by Linyuan Lu, Linyuan Lü, Lu, Linyuan +2
Computer Science · Mathematics · #Algorithms and Data Compression #Limits and Structures in Graph Theory #Stochastic processes and statistical mechanics #math.CO #msc:05A20 #msc:05C30 #msc:05C80

paper · pdf · doi:10.48550/arxiv.1506.08382

9 pages

arxiv created 2015/06/28 · arxiv updated 2015/06/30

Abstract

Let f(m,c)=∑_k=0^∞ (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.

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