Racah polynomials

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 harvtxt|Wilson|1978 and are given by:p_n(x(x+gamma+delta+1)) = {}_4F_3left [egin{matrix} -n &n+alpha+eta+1&-x&x+gamma+delta+1\alpha+1&gamma+1&eta+delta+1\ end{matrix};1 ight]

harvtxt|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) = {}_4phi_3left [egin{matrix} q^{-n} &abq^{n+1}&q^{-x}&q^{x+1}cd\aq&bdq&cq\ end{matrix};q;q ight] They are sometimes given with changes of variables as:W_n(x;a,b,c,N;q) = {}_4phi_3left [egin{matrix} q^{-n} &abq^{n+1}&q^{-x}&cq^{x-n}\aq&bcq&q^{-N}\ end{matrix};q;q ight]

References

*Citation | last1=Askey | first1=Richard | last2=Wilson | first2=James | title=A set of orthogonal polynomials that generalize the Racah coefficients or 6-j symbols | doi=10.1137/0510092 | id=MathSciNet | id = 541097 | year=1979 | journal=SIAM Journal on Mathematical Analysis | issn=0036-1410 | volume=10 | issue=5 | pages=1008–1016
*citation|first=J.|last= Wilson|title= Hypergeometric series recurrence relations and some new orthogonal functions|series= Ph.D. thesis|publisher= Univ. Wisconsin, Madison|year= 1978


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