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Racah polynomials

From Wikipedia, the free encyclopedia

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