# Analytic ordinary generating function

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

An ordinary power series A(x)=sum over n≥0 of a_n*x^n analytically encodes a sequence on an explicitly declared convergence domain.

mko-analytic-ordinary-generating-function · definition · v0.10.0



## 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

[A Fibonacci-type OGF with initial boundary terms](https://ocme.evemisslab.com/en/derivations/derivation-mko-analytic-ordinary-generating-function-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Recurrence relation](https://ocme.evemisslab.com/en/mko/mko-recurrence-relation/) — Derive an OGF from the complete recurrence.
- [Radius of convergence of a power series](https://ocme.evemisslab.com/en/mko/mko-power-series-radius-of-convergence/) — State the analytic domain.
- [Sequence](https://ocme.evemisslab.com/en/mko/mko-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.

[Canonical MKO JSON](https://ocme.evemisslab.com/data/mko/analytic-ordinary-generating-function.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-analytic-ordinary-generating-function/index.md)