MATH 245 CH14 - DISCRETE MATHEMATICS W W L CHEN c W W L...

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DISCRETE MATHEMATICS W W L CHEN c ± W W L Chen, 1992, 2008. This chapter is available free to all individuals, on the understanding that it is not to be used for financial gain, and may be downloaded and/or photocopied, with or without permission from the author. However, this document may not be kept on any information storage and retrieval system without permission from the author, unless such system is not accessible to any individuals other than its owners. Chapter 14 GENERATING FUNCTIONS 14.1. Introduction Definition. By the generating function of a sequence a 0 ,a 1 ,a 2 ,a 3 ,..., we mean the formal power series a 0 + a 1 X + a 2 X 2 + a 3 X 3 + ... = X n =0 a n X n . Example 14.1.1. The generating function of the sequence ± k 0 ² , ± k 1 ² ,..., ± k k ² , 0 , 0 , 0 , 0 ,... is given by ± k 0 ² + ± k 1 ² X + ... + ± k k ² X k . This is equal to (1 + X ) k by the Binomial theorem. Example 14.1.2. The generating function of the sequence 1 ,..., 1 | {z } k , 0 , 0 , 0 , 0 ,... is given by 1 + X + X 2 + X 3 + ... + X k - 1 = 1 - X k 1 - X . Chapter 14 : Generating Functions page 1 of 6
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Discrete Mathematics c ± W W L Chen, 1992, 2008 Example 14.1.3. The generating function of the sequence 1 , 1 , 1 , 1 ,... is given by 1 + X + X 2 + X 3 + ... = X n =0 X n = 1 1 - X . Example 14.1.4. The generating function of the sequence 2 , 4 , 1 , 1 , 1 , 1 ,... is given by 2 + 4 X + X 2 + X 3 + X 4 + X 5 ... = (1 + 3 X ) + (1 + X + X 2 + X 3 + ... ) = (1 + 3 X ) + X n =0 X n = 1 + 3 X + 1 1 - X . 14.2. Some Simple Observations The idea used in the Example 14.1.4 can be generalized as follows. PROPOSITION 14A. Suppose that the sequences a 0 ,a 1 ,a 2 ,a 3 ,... and b 0 ,b 1 ,b 2 ,b 3 ,...
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MATH 245 CH14 - DISCRETE MATHEMATICS W W L CHEN c W W L...

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