TL;DR

A formal power series \( G(x) = \sum_{n=0}^{\infty} a_n x^n \) encoding a sequence \( \{a_n\} \). Operations on the power series correspond to operations on the sequence.

By Valenke Exam Prep Team·Last updated 2026-06-03

Generating Function

A formal power series \( G(x) = \sum_{n=0}^{\infty} a_n x^n \) encoding a sequence \( \{a_n\} \). Operations on the power series correspond to operations on the sequence.

Why it matters for interviews

Generating functions transform combinatorial problems into algebraic ones. They solve recurrence relations, compute probability distributions, and appear in advanced interview problems on counting and probability.

Definition and Mathematical Foundation

A formal power series \( G(x) = \sum_{n=0}^{\infty} a_n x^n \) encoding a sequence \( \{a_n\} \). Operations on the power series correspond to operations on the sequence.

Application in Quantitative Finance

Related Terms

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Frequently Asked Questions

What is the difference between ordinary and exponential generating functions?▼

OGF: \( \sum a_n x^n \) (for combinations/selections). EGF: \( \sum a_n x^n/n! \) (for permutations/labeled structures). The choice depends on whether the problem involves labeled or unlabeled objects.

How do generating functions solve recurrences?▼

Multiply the recurrence by \( x^n \), sum over n to get an equation for \( G(x) \), solve algebraically, then extract coefficients via partial fractions or Taylor expansion.

What is the probability generating function?▼

For a non-negative integer-valued random variable: \( G(s) = E[s^X] = \sum P(X=k) s^k \). The mean is \( G'(1) \), and the factorial moments are higher derivatives at 1.