generating_functions4 - Power series (a0 , a1 , a2 ,) :...

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1 COS 341 Discrete Mathematics Generating Functions 2 Power series 2 01 2 Infinite series of the form aa x a x ++ + " 2 1 1 1 xx x =++ + " Function contains all the information about series Differentiate k times and substitute x=0 , we get k! times coefficient of x k Series converges for x in the interval (-1,1) 1 Taylor series of the function at 0 1- x x = 3 Power series 012 11 0 () ( , , , ) :sequence of real numbers || For any number ( , ), the series ( ) converges Values of a(x) in arbitrarily small neighborhod of 0 uniquely determine ( , , , ) (0 n n KK i i i n n aaa aK x ax a x a a = ∈− = = ) ! n 4 Generating functions 2 0 ( , , , ) :sequence of real numbers of this sequence is the power serie Gene s rating function i i i a = =
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5 Generating function toolkit: Generalized binomial theorem (1 ) (2 ) ( 1 ) ! r rr r r k k k  −− + =    rrrr ,,,, 0123 is the generating function for the sequence ) r x +       23 r r 01 2 3 The power series always converges for all ||1 xx x x ++ + +        < 6 Negative binomial coefficients ? 11 ) ) 1 kk k r k r + +  = − = 2 2 n-1 n n+1 n+k-1 n-1 n-1 n-1 n-1 1 ) 1 1 1 k n x x x + + + ++ +      =      = "… " 7 Operations on power series • Addition • Multiplication by fixed real number • Shifting the sequence to the right • Shifting to the left 00 has generating function ( , , ) () ab a xb x + has generating functi (,, ) ( on ) aa a x αα α N has generating fu (0, 0 ncti ,, on ,) ( ) n n xax × …… 1 0 1 has generating f ( unction ) (, ,) k i i i n ax a x x = + −⋅ 8 • Substituting α x for x • Substitute x n for x NN 012 has ge (, 0 , 0 , nerati ,0, 0, ng function )( ) n nn aaa a x
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generating_functions4 - Power series (a0 , a1 , a2 ,) :...

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