Racah polynomials
In mathematics, Racah polynomials are orthogonal polynomials named after Giulio Racah, as their orthogonality relations are equivalent to his orthogonality relations for Racah coefficients.
The Racah polynomials were first defined by Wilson (1978) and are given by
Orthogonality
- when ,
- where is the Racah polynomial,
- is the Kronecker delta function and the weight functions are
- and
- is the Pochhammer symbol.
Rodrigues-type formula
- where is the backward difference operator,
Generating functions
There are three generating functions for
- when or
- when or
- when or
Connection formula for Wilson polynomials
When
- where are Wilson polynomials.
q-analog
Askey & Wilson (1979) introduced the q-Racah polynomials defined in terms of basic hypergeometric functions by
They are sometimes given with changes of variables as