13/11/29 20:03:24.93
>>403
与式を Q_m とおく。母関数は
Σ[m=0,∞) Q_m/m!･s^m
= Σ[m=0,∞) Σ[n=1,∞) P[n+m-1,m]/m! (1-r)^(n-1)･s^m
= Σ[m=0,∞) Σ[n=1,∞) C[n+m-1,m] (1-r)^(n-1)･s^m
= Σ[n '=0,∞) Σ[m=0,n '] C[n ',m] (1-r)^n '-m･s^m
= Σ[n '=0,∞) (1-r+s)^n '
= 1/(r-s)
= (1/r)/{1-(s/r)}
= (1/r)Σ[m=0,∞) (s/r)^m,
∴ Q_m = m!/r^(m+1),