structural:expansion, structural:summarization
Convolution (Cauchy products)

A discrete convolution of the terms in two formal power series turns a product of generating functions into a generating function enumerating a convolved sum of the original sequence terms (see Cauchy product).

Multiplication of generating functions, or convolution of their underlying sequences, can correspond to a notion of independent events in certain counting and probability scenarios. For example, if we adopt the notational convention that the probability generating function, or pgf, of a random variable Z is denoted by G (z), then we can show that for any two random variables G X + Y (z) = G X (z) G Y (z), {\displaystyle G_{X+Y}(z)=G_{X}(z)G_{Y}(z)\,,} if X and Y are independent. Similarly, the number of ways to pay n ≥ 0 cents in coin denominations of values in the set {1, 5, 10, 25, 50} (i.e., in pennies, nickels, dimes, quarters, and half dollars, respectively) is generated by the product C (z) = 1 1 − z 1 1 − z 5 1 1 − z 10 1 1 − z 25 1 1 − z 50, {\displaystyle C(z)={\frac {1}{1-z}}{\frac {1}{1-z^{5}}}{\frac {1}{1-z^{10}}}{\frac {1}{1-z^{25}}}{\frac {1}{1-z^{50}}},} and moreover, if we allow the n cents to be paid in coins of any positive integer denomination, we arrive at the generating for the number of such combinations of change being generated by the partition function generating function expanded by the infinite q -Pochhammer symbol product of ∏ n = 1 ∞ (1 − z n) − 1. {\displaystyle \prod _{n=1}^{\infty }\left(1-z^{n}\right)^{-1}\,.}