The distributions of some quantities for Erlang(2) risk models

From Dickson and Li (2010), the inverse of (r 1 + ) 1=2 is a function V 1 (t)given byV 1 (t) = e (+c)t 1X m t m 1 m=2 (ct) (m+1)=2m+12 m!+ 1 2m=0and the inverse of (r 2 + ) 1=2 is a function W 1 (t) given byW 1 (t) = e (+c)t21X ( ) m t m 1 m=2 (ct) (m+1)=2m+1m!+ 1 2m=0meaning that the inverse of (r 1 + ) 1=2 (r 2 + ) 1=2 is a function, say g(t),given by1Xg(t) = n e (+c)t t 3n+1wheren=0 n = 2n+1 n+1=2 c n+1(2n + 1)! (n + 2) :Hence the inverse of (4.5) is the convolution of g with a function h, wherefrom (4.4)1Xh(t) = h m e (+c)t t 3m+1wherem=0h m = 2m+1 m+3=2 c m+1:(2m + 2)! m!The Laplace transform (with parameter s) of the product of the m-th term ofh(t) and the n-th term of g(t) isgivingwhere l =h m n(3m + 2) (3n + 2)( + c + s) 3(m+n)+4h g(t) =1Xl=0 le (+c)t t 3l+3(3l + 4)lXh m l m (3m + 2) (3 (l m) + 2)m=0lX= l+2 2l+2 c l+2 (3m + 1)! (3(l m) + 1)!(2m + 2)! m! (2(l m) + 1)!(l m + 1)!m=010