By Arto Salomaa
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Extra resources for Computation and automata
M (34) In other words, emllg (m ,a) is the egf, over n, of mn f (m ,n). It is as simple as it is ingenious. Writing explicitly a gf2 for g (m ,a) g(m,a)= Laiai, (35) i~ we have k mk-I emllg (m ,a) = ~ ck a k , where c - ~ a - - kJ k -lkJ=o 1 (k-l)! ' (36) k~ and equation (34) then yields n! n n1 f(m,n)=c n - n = LalI ' m 1=0 m (37) where n 1 is the descending factorial: n1 = n(n-l) ... /(n-l)!. (In some of the literature this quantity is denoted by (n )1') f Given g (m ,a), equations (35) and (37) provide a complete expansion for (m ,n).
The operations of addition, subtraction, multiplication, differentiation and integration are defined in the intuitive way yielding, inter alia (2) An equally valid view holds z to be a variable in the complex plane, and then the question of convergence of the defining series arises. The analytic properties of gf's then frequently yield information that is quite hard to come by otherwise. 5. There is never the need to choose and settle on an a priori interpretation. When gf's are used in the sequel we shall vacillate freely between the two points of view.
1-z (1-z)2 Equation (5) with bi =1 can thus be iterated, and one obtains sums of sums to any desired degree. Let us use the notation Sn ai = S [sn-I ai ], that is SOai = ai S lai =: Sai = ao + a 1 + ... + ai (9) S2ai =S(ao+ al + ... +ai) = (i+l)a o + ia 1 + ... + 2ai_1 + ai = i L (k+1)ai_k ' k=O and so on. Experimenting with n = 2, 3 suggests the following expression for S n ai ' which we prove by mathematical induction: S n ai = Li (n+k-1) n-1 ai-k' k=O For n = 2 equation (10) reduces to the second line in equation (9).
Computation and automata by Arto Salomaa