gen-define-generating-functions-and-give-one-short-worked-exampl-423e

Define generating functions and give one short worked example.

# Generating Functions ## Definition A **generating function** is a formal power series used to encode a sequence of numbers. For a sequence $(a_0, a_1, a_2, \ldots)$, the generating function is: $$G(x) = a_0 + a_1x + a_2x^2 + a_3x^3 + \cdots = \sum_{n=0}^{\infty} a_n x^n$$ The sequence values become coefficients, and the generating function serves as a tool for solving combinatorial problems, recurrence relations, and finding closed forms. ## Worked Example **Problem:** Find a closed form for the Fibonacci sequence using generating functions. **Solution:** Let $F(x) = \sum_{n=0}^{\infty} f_n x^n$ where $f_n$ is the $n$-th Fibonacci number with $f_0 = 0, f_1 = 1$, and $f_n = f_{n-1} + f_{n-2}$. Starting with the recurrence: $$F(x) = f_0 + f_1x + \sum_{n=2}^{\infty} f_n x^n$$ $$F(x) = x + \sum_{n=2}^{\infty} (f_{n-1} + f_{n-2}) x^n$$ $$F(x) = x + x\sum_{n=2}^{\infty} f_{n-1} x^{n-1} + x^2\sum_{n=2}^{\infty} f_{n-2} x^{n-2}$$ $$F(x) = x + xF(x) + x^2F(x)$$ Solving for $F(x)$: $$F(x)(1 - x - x^2) = x$$ $$F(x) = \frac{x}{1 - x - x^2}$$ This closed form can then be used to extract Fibonacci numbers or derive Binet's formula.