H. Koo
Northwood High School, Irvine, CA, United State
Abstract
Generating functions are algebraic objects that encode sequences as coe!cients of power series, allowing problems about counting and recurrence relations to be translated into problems about algebraic manipulation. This paper introduces several major types of generating functions—ordinary, exponential, Fibonacci, and Dirichlet generating functions—and explores how each can be used to solve combinatorial and number-theoretic problems. Through examples involving counting, Pascal’s triangle, the hockey-stick identity, Fibonacci numbers, the binomial theorem summation identity, and Euler’s totient function, the paper shows how generating functions unify ideas from combinatorics, algebra, and number theory. These examples illustrate how generating functions turn recursive or case-based problems into systematic coe!cient-extraction problems and provide elegant proofs of important identities
