---
tag: P6JYNDX5DR
kind: concept
name: Racah polynomials
msc: [33C45, 33C80]
sources:
  - url: https://en.wikipedia.org/wiki/Racah_polynomials
same_as:
  wikidata: Q3907354
  wikipedia: Racah polynomials
---

## Summary

The Racah polynomials are a finite family of orthogonal polynomials, given by terminating balanced ${}_4F_3$ hypergeometric series.
They are polynomials in the quadratic variable $\lambda(x) = x(x + \gamma + \delta + 1)$, orthogonal for a weight on the points $x = 0, 1, \dots, N$.
They sit at the top of the [[Askey scheme|RR2S05C09B]] together with the [[Wilson polynomials|1CPF7SWQ3S]].
They are named after Giulio Racah because the [[6-j symbols|D17XX5N9VF]] of quantum angular momentum can be written in terms of them.

## Explanations

### L1 human

In the Askey scheme, families of orthogonal polynomials are arranged by how many parameters they have.
The Racah polynomials are the discrete family with the most parameters, four of them.
The other discrete families, such as the Hahn, Krawtchouk and Charlier polynomials, arise from them by letting parameters go to limits.
In physics they appear when three quantum angular momenta are added in two different orders.