---
tag: NWKX329T01
kind: concept
name: Holonomic sequence
msc: [39A06, 33F10, 05A15]
sources:
  - url: https://en.wikipedia.org/wiki/Holonomic_function
---

## Summary

A holonomic sequence, also called a P-recursive sequence, is a sequence that satisfies a [[linear recurrence with polynomial coefficients|K6QV4WSTXX]].
A sequence is holonomic exactly when its generating function is a [[holonomic function|7H44JD9NQD]].
The class is closed under sums, termwise products and many other operations, which makes it the basis of automatic proofs of identities for sums.
It contains the solutions of linear recurrences with constant coefficients, the factorials, the Catalan numbers and the harmonic numbers.

## Explanations

### L1 human

Some sequences grow in complicated ways, but still obey a simple linear rule whose coefficients are polynomials in the position $n$.
The factorials obey $u_{n+1} = (n+1) u_n$, so they are holonomic.
Such a rule together with a few starting values is a complete, finite description of the sequence.
A computer can store it, add two such sequences, or check an identity by comparing rules.