508 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 



the (unlimited) common group, namely 



{a -{- m -\- n)f(m, n) — (w + l)f(m + 1, n) 



— (n + l)/(m, n + 1) — af(in — 1, n) = 



(1-2) « 



(a -{- X -{- n)j{x, n) — af{x, n — 1) \ 



- (n -\- l)f(x, n + 1) - af(x - 1, n) = 



and 



/(m, n) = 0, m < or n < or m > x, 



factorial moment generating function recurrences may be found and 

 solved. 



With m fixed, factorial moments of n are defined by 



M(fc)(m) = E {n)kf{m, n) (1.3) 



n=0 



or alternatively by the factorial moment exponential generating function 

 M{m, = Z MUm)t'/k\ = £ (1 + 07K n) (1.4) ] 



fc=0 n=0 I 



In (1.3), {n)k = n{n — 1) • • • (n — /c + 1) is the usual notation for a \ 

 falling factorial. 



Using (1.4) in equations (1.2), and for brevity D = d/dt, it is found 

 that 



a^ m ^- tD)M{in, t) - (m + l)M{m + 1, t) 



- aM(m - l,t) = (1.5) 



(x - at -\- tD)M{x, t) - aM{x - \,t) = 



which correspond (by equating powers of t) to the factorial moment re- 

 currences 



{a-\- m^ k)M^kM) - (m + l)Ma)(w + 1) 



- ailf (fc)(m - 1) = (1.6) 



(x + k)M(k)(x) - akM^k-i)ix) - aMik)(x - 1) = 



Notice that the first of (1.6) is a recurrence in m, which suggests (fol- 

 lowing Molina) introducing a new generating function defined by 



Gdu) = T.M^k){m)u'^ (1.7) 



