exponential_generating_functions4

exponential_generating_functions4 - Exponential Generating...

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1 COS 341 Discrete Mathematics Exponential Generating Functions 2 Generating Functions 2 0 01 ( , , , ) :sequence of real numbers of this sequence is the power serie Gene s rating Function () i i i aa a xx = = Ordinary Ordinary 3 Exponential Generating Functions 2 0 Exponential Generating func ( , , , ) :sequence of real numbers of this sequence is the power se ries ! tion i i i a ax x i a = = 4 Exponential generating function examples What is the generating function for the sequenc (1,1,1 e ,1, ? ) 23 0 1 1 !1 ! 2 ! 3 ! ix i xx x x e i = =+ + + + = " What is the generating function for the sequenc (1,2,4 e ,8, ? ) 2 0 22 ( 2 ) ( 2 ) 1 ! 2 ! 3 ! i i x x e i = =+ + + + = "
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5 Operations on exponential generating functions • Addition • Multiplication by fixed real number • Shifting the sequence to the right • Shifting to the left 00 11 has generating function ( , , ) () ab a xb x ++ + 01 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 kk n ax a x x = + −⋅ 6 • 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 × × 2 01 2 has generating funct (, , ion ) aa a α 7 • Differentiation • Integration • Multiplication of generating functions 123 has generating function 2, 3 ) (or ) '( ) d a x a x dx 23 0 has generating function (0, , , ) ( ) x f t d t 0 0 n n n n nk n k k bx cx ca b ∞∞ == = = ⋅⋅ = = ∑∑ 8 Differentiation is the exponential generating function f ( , , o , r ) a a a Differentiation is equivalent to shifting the sequence to the left 0 !
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exponential_generating_functions4 - Exponential Generating...

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