Difference between revisions of "Generating function"
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The reason to go to such lengths is that our above polynomial is equal to <math>(1+x)^n</math> (which is clearly seen due to the [[Binomial Theorem]]). By using this equation, we can rapidly uncover identities such as <math>{n \choose 0}+{n \choose 1}+...+{n \choose n}=2^n</math>(let <math>{x}=1</math>), also <math>{n \choose 1}+{n \choose 3}+\cdots={n \choose 0}+{n \choose 2}+\cdots</math>. | The reason to go to such lengths is that our above polynomial is equal to <math>(1+x)^n</math> (which is clearly seen due to the [[Binomial Theorem]]). By using this equation, we can rapidly uncover identities such as <math>{n \choose 0}+{n \choose 1}+...+{n \choose n}=2^n</math>(let <math>{x}=1</math>), also <math>{n \choose 1}+{n \choose 3}+\cdots={n \choose 0}+{n \choose 2}+\cdots</math>. | ||
+ | |||
+ | == Convolutions == | ||
+ | Suppose we have the sequences <math>a_1, a_2, a_3, ...</math> and <math>b_1, b_2, b_3, ...</math>. We can create a new sequence, called the convolution of <math>a</math> and <math>b</math>, defined by <math>c_n = a_0b_n + a_1b_{n-1} + ... + a_nb_0</math>. Generating functions allow us to represent the convolution of two sequences as the product of two power series. If <math>A</math> is the generating function for <math>a</math> and <math>B</math> is the generating function for <math>b</math>, then the generating function for <math>c</math> is <math>AB</math>. | ||
=== See also === | === See also === |
Revision as of 22:02, 15 November 2007
This is an AoPSWiki Word of the Week for Nov 15-21 |
The idea behind generating functions is to create a power series whose coefficients, , give the terms of a sequence which of interest. Therefore the power series (i.e. the generating function) is and the sequence is .
Simple Example
If we let , then we have .
This function can be described as the number of ways we can get heads when flipping different coins.
The reason to go to such lengths is that our above polynomial is equal to (which is clearly seen due to the Binomial Theorem). By using this equation, we can rapidly uncover identities such as (let ), also .
Convolutions
Suppose we have the sequences and . We can create a new sequence, called the convolution of and , defined by . Generating functions allow us to represent the convolution of two sequences as the product of two power series. If is the generating function for and is the generating function for , then the generating function for is .
See also
- Combinatorics
- Polynomials
- Series
- generatingfunctionology a PDF version