Generating function

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 is 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 $a_0, a_1, a_2, ...$ and $b_0, b_1, b_2, ...$. 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 $AB$.

Simple Exercises

1. There are three baskets on the ground: one has 2 purple eggs, one has 2 green eggs, and one has 3 white eggs. Eggs of the same color are indistinguishable. In how many ways can I choose 4 eggs from the baskets?

Solution for problem 1: The generating function for the first basket is $1+x+x^2$, since there is one way to choose 0 purple eggs (do nothing), 1 way to choose 1 purple egg (since eggs are indistinguishable), and 1 way to choose 2 purple eggs. In a similar fashion the generating function for the green egg basket is $1+x+x^2$ and the generating function for the white egg basket is $1+x+x^2+x^3$. Now to find the number of ways to pick eggs from multiple baskets, just simply multiply the functions together, getting $1+3x+6x^2+8x^3+8x^4+6x^5+3x^6+x^7$. We want the number of ways to choose 4 eggs, so we just need to look at the coefficient of $x^4$ and see that there are 8 ways to choose 4 eggs.

See also

External Link

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