Generating function

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, $c_0, c_1, c_2, \ldots$ , give the terms of a sequence which of interest. Therefore the power series (i.e. the generating function) is $c_0 + c_1 x + c_2 x^2 + \cdots$ and the sequence is $c_0, c_1, c_2,\ldots$ .

Simple Example

If we let $A(k)={n \choose k}$ , then we have ${n \choose 0}+{n \choose 1}x + {n \choose 2}x^2+\cdots+$ ${n \choose n}x^n$ .

This function can be described as the number of ways we can get ${k}$ heads when flipping $n$ different coins.

The reason to go to such lengths is that our above polynomial is equal to $(1+x)^n$ (which is clearly seen due to the Binomial Theorem). By using this equation, we can rapidly uncover identities such as ${n \choose 0}+{n \choose 1}+...+{n \choose n}=2^n$ (let ${x}=1$ ), also ${n \choose 1}+{n \choose 3}+\cdots={n \choose 0}+{n \choose 2}+\cdots$ .

Convolutions

Suppose we have the sequences $a_1, a_2, a_3, ...$ and $b_1, b_2, b_3, ...$ . We can create a new sequence, called the convolution of $a$ and $b$ , defined by $c_n = a_0b_n + a_1b_{n-1} + ... + a_nb_0$ . Generating functions allow us to represent the convolution of two sequences as the product of two power series. If $A$ is the generating function for $a$ and $B$ is the generating function for $b$ , then the generating function for $c$ is $AB$ .

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Generating function

Simple Example

Convolutions

See also