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

$p_n(x(x+\gamma+\delta+1)) = {}_4F_3\left[\begin{matrix} -n &n+\alpha+\beta+1&-x&x+\gamma+\delta+1\\ \alpha+1&\gamma+1&\beta+\delta+1\\ \end{matrix};1\right].$

Askey & Wilson (1979) introduced the q-Racah polynomials defined in terms of basic hypergeometric functions by

$p_n(q^{-x}+q^{x+1}cd;a,b,c,d;q) = {}_4\phi_3\left[\begin{matrix} q^{-n} &abq^{n+1}&q^{-x}&q^{x+1}cd\\ aq&bdq&cq\\ \end{matrix};q;q\right].$

They are sometimes given with changes of variables as

$W_n(x;a,b,c,N;q) = {}_4\phi_3\left[\begin{matrix} q^{-n} &abq^{n+1}&q^{-x}&cq^{x-n}\\ aq&bcq&q^{-N}\\ \end{matrix};q;q\right].$

References

• Askey, Richard; Wilson, James (1979), "A set of orthogonal polynomials that generalize the Racah coefficients or 6-j symbols", SIAM Journal on Mathematical Analysis 10 (5): 1008–1016, doi:10.1137/0510092, ISSN 0036-1410, MR 541097
• Wilson, J. (1978), Hypergeometric series recurrence relations and some new orthogonal functions, Ph.D. thesis, Univ. Wisconsin, Madison