  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 $c$.
