510 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 



The first of these leads to 



cr/^(0) + (7,(1) + • • • + cr,(x) = ak+i(x) (1.14) J 

 and the last is useful in the form j 



kak+i{m) = {m + k - a)ak{m) + 0(rt_i(m) (1.15) ■- 



Also, the first along with ao(m) = a" /m\ leads to a simple calculation ; 

 procedure, as Kosten has noticed. 



By (1.11) the factorial moments are now completely determined ex- 

 cept for il/(A-)(0). To determine the latter, the second of (1.6) and the 

 normalizing equation 



X 



E M,{m) = 1 (1.16) 



are available. 



Thus from the second of (1 .6) 



[(:r + k)<r,{x) - mu{x - l)].^/(A-)(0) = a/v(r,_i(.c)M(,_i)(0) (1.17) 



Also 



{x + k)ak{x) — acTkix — 1) 



= (x -\- k - a)ak(x) + a[(Xk{x) - (Tk(.x — 1)] 



= (x -\- k - a)(Tk{x) + a<Tk-i{x) 



= /t'o-fc+i^r), 



the last step by (1.15). Hence 



(Tk-l{x) 



MaM = a "-^=^ Ma-iM (1.18) 



<rk+i{x) 



and by iteration 



^k (7i(x)(roix) 



MaAO) = a' "^7" "7, Mo(0) (1.19) 



From (l.ll) and (1.16), and in the last step (1.14), 



t.M,(m) = i: il/o(0)cro(7n) = ilfo(0)cri(a;) = 1 (1.20) 



Hence finally 



Ma)(m) = Ma-M<rk{ni) 



, a,(x)ak(m) (1.21) 



= a 



(Tki.i{.r)<^k{x) 



