---
tag: RR2S05C09B
kind: concept
name: Askey scheme
msc: [33C45]
sources:
  - url: https://en.wikipedia.org/wiki/Askey_scheme
same_as:
  wikidata: Q4807189
  wikipedia: Askey scheme
---

## Summary

The Askey scheme is a chart that organises the hypergeometric orthogonal polynomials into levels connected by limit relations.
At the top are the [[Wilson|1CPF7SWQ3S]] and [[Racah polynomials|P6JYNDX5DR]], given by ${}_4F_3$ series; at the bottom are the [[Hermite polynomials|MGACY98E9H]].
Each arrow says that one family is a limit of another as a parameter tends to a special value or to infinity.
It shows the classical families of Jacobi, Laguerre and Hermite as degenerations of a few very general ones.

## Explanations

### L1 human

There are many named families of orthogonal polynomials, and at first they look unrelated.
The Askey scheme is a family tree for them.
Families with more parameters are higher up, and moving down an arrow means fixing or removing a parameter by a limit.
For example, the [[Krawtchouk polynomials|CJGKW80KBR]] for $N$ coin tosses with probability $a/N$ become the [[Charlier polynomials|6KH8KVYS3R]] as $N \to \infty$, just as the binomial distribution becomes the Poisson distribution.