Analytic ordinary generating function

Mathematical statement

Coefficient extraction [x^n]A=a_n is the encoding obligation. A finite partial sum uses only its declared prefix. For a_n=r^n, A(x)=1/(1−r*x) holds analytically only when |r*x|<1. For a_n=a_(n−1)+a_(n−2) with explicit a0,a1, the OGF is (a0+(a1−a0)*x)/(1−x−x^2); initial boundary terms must not be omitted. A formal power-series identity is not automatically an analytic equality. Do not evaluate outside convergence, determine a whole OGF/radius from an unknown finite prefix or infer unproved termwise differentiation from the word analytic.

Representative formula (exact TeX)

A(x)=G(a,x)

Assumptions

• Exact families use rational coefficients/points with an explicit sufficient convergence domain.
• The recurrence family is the homogeneous order-two Fibonacci-type law with both initial values.

Symbol scope

• A — analytic ordinary generating function
• x — evaluation variable
• G — ordinary-generating operator
• a — coefficient sequence

Intuition, definition, and misconceptions

The sequence and coefficient extraction align one-to-one.

A geometric closed form needs its convergence domain.

Summing a recurrence after multiplication by x^n produces explicit initial boundary terms.

Do not substitute an arbitrary x into a formal identity as a numerical equality or infer a complete generating function from an unknown prefix.

Adopted derivation guide

The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.

Prerequisites

• Recurrence relation — Derive an OGF from the complete recurrence.
• Radius of convergence of a power series — State the analytic domain.
• Sequence — Coefficients are indexed data.

Verification boundary

This node uses explicitly declared ordinary analytic instances, not automatically exponential generating functions or formal-only algebra.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.

Next obligation

Add generating-function operations and asymptotic coefficient analysis.

Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.