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

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 [1]
when  ,
where   is the Racah polynomial,
 
  is the Kronecker delta function and the weight functions are
 
and
 
  is the Pochhammer symbol.

Rodrigues-type formula

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 [2]
where   is the backward difference operator,
 

Generating functions

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There are three generating functions for  

when  or 
 
 
when  or 
 
 
when  or 
 
 

Connection formula for Wilson polynomials

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When  

 
where   are Wilson polynomials.

q-analog

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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

 

References

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  1. ^ Koornwinder, Tom H.; Wong, Roderick S. C.; Koekoek, Roelof; Swarttouw, René F. (2010), "Wilson Class: Definitions", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248.
  2. ^ Koekoek, Roelof; Swarttouw, René F. (1998), The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue