This preview shows pages 1–5. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: July 30, 1996 1 GENERATING FUNCTIONS Solve an infinite number of related problems in one swoop. *Code the problems, manipulate the code, then decode the answer! Really an algebraic concept but can be extended to analytic basis for interesting results. (i) Ordinary Generating Functions {a , a 1 , , a k , } sequence where the kth term is the solution of some problem, for every k. Create the object (formal power series) k=0 k k a x where x k is like a placeholder for a k . This looks like an analytic power series but its NOT (not yet, anyway). Rules of Operation: just do what comes naturally. ( 29 a x b x = a b x k k k k k k k ( 29 ( 29 a x b x = c x k k k k k k where c k = j=0 k j k j a b Examples (I) a k = 1 , 0 k n a k = 0 k > n k=0 k k n n+1 a x = 1 + x + _ + x = 1  x 1  x why is last equality true? Because ....... (1 + x + + x n )(1  x) = 1 + x + + x n =  x  _  x  x 1  x n n+1 n+1 (ii) a k = 1 2200 k. July 30, 1996 2 k=0 k x = 1 1  x (1  x) x x x x x = 1 k=0 k k k k k+1 k k k 1 k = = NOTE: You dont need anything about convergence! At the same time, you shouldnt think of x as a variable into which you substitute values (not yet, anyway) but soon it will be OK). (iii) If we have the o.g.f., we can find the sequence: e.g. Suppose the o.g.f. is (1 + x) n then the sequence is found as follows: (1 + x ) = n k x n k k Thus, a k = n k (note that a k = 0 for k > n). Exponential Generating Function {a , a 1 , , a k , } k k k a k! x . k k k k k k k k k a k! x b k! x = c k! x , where k j=0 k j k j c = j k a b c = a b k j=0 k j k j k j=0 k j k c k! = a j! b (k j)! Examples (i) a k = 1, 0 k n; a k = 0 for k > n July 30, 1996 3 k=0 n k x k! (ii) a k = 1 2200 k k k x x k! = e (iii) a k = n k ( number of kperms of an nset) (iv) If we have the e.g.f. sin x then sin x = k k 2 k+1 (1) (2 k+ 1)! x so a 2k + 1 = (1) k k a 2k = 0 k Some Generating Function Manipulations Suppose A(x) = k k k k k a x , B(x) = x  Then A(x) B)(x) = k k k d x where k j=0 k j j=0 k j c = a 1 = a  Similarly, A 2 (x) = k k k d x where k j=0 k j k j d = a a e.g. a k = n k then 2 n n = n k k=0 n 2 July 30, 1996 4 Also, 1 x [A(x)  a ] = a x k k+1 k , which is the o.g.f. for the sequence {a 1 ,a 2 , ,a k , } which is the original sequence shifted one place to the left (and the first term dropped off)....
View Full
Document
 Fall '06
 miller

Click to edit the document details