COINCIDENCES IN POISSON PATTERNS 1015 



2.^ Laplace Transform of Fn{L) 



To analyze the system of equations which is given by relations (2-1) 

 through (2-6) , we introduce the generating functions 



f{L,t) = ZFAL)^, 



and 



p(L,t) = f:PAL)f. 



If L > 5, we obtain from (2-3) and (2-4) the relation 



e^'-fiL, = 1 + f v(w, Oe'^X dw, (2-7) 



and from (2-5) and (2-6) the relation (again if L > 8) 



'p{L, = \t C p{w, Oe'" dw + e^''-''f{L - 5, 0- (2-8) 



XL 



e 



If we differentiate (2-7) and (2-8) with respect to L, and then apply 

 (2-7) differentiated to simphfy the last terms of (2-8) differentiated, we 

 obtain, still only f or L > 8, 



f'iL, t) + X/(L, = ^P(L, i), (2-9) 



p'(L, + X(l - t)p{L, t) = \e-^\l - t)p{L - 8, t). (2-10) 



It is easy to check from (2-1) and (2-2) that if L ^ 5, then 



/T ,N -XL (1-0 



p{L, t) = e 

 and 



and hence (2-9) is valid for all L, but the left side of (2-10) vanishes if 

 L ^ 8. Hence we may take Laplace transforms of (2-9) and (2-10). If 

 we define 



and 



Ais,t) = r f{L, t)e-'' dL, 

 Jo 



. B{s, = [ p{L, Oe-^" dL, 



JO 



