Racah polynomials

Orthogonality

:\sum_{y=0}^N\operatorname{R}n(x;\alpha,\beta,\gamma,\delta) \operatorname{R}m(x;\alpha,\beta,\gamma,\delta)\frac{\gamma+\delta+1+2y}{\gamma+\delta+1+y} \omega_y=h_n\operatorname{\delta}{n,m}, :when \alpha+1=-N, :where \operatorname{R} is the Racah polynomial, :x=y(y+\gamma+\delta+1), :\operatorname{\delta}{n,m} is the Kronecker delta function and the weight functions are :\omega_y=\frac{(\alpha+1)_y(\beta+\delta+1)_y(\gamma+1)_y(\gamma+\delta+2)_y}{(-\alpha+\gamma+\delta+1)_y(-\beta+\gamma+1)_y(\delta+1)_yy!}, :and :h_n=\frac{(-\beta)_N(\gamma+\delta+1)_N}{(-\beta+\gamma+1)_N(\delta+1)_N}\frac{(n+\alpha+\beta+1)nn!}{(\alpha+\beta+2){2n}}\frac{(\alpha+\delta-\gamma+1)_n(\alpha-\delta+1)_n(\beta+1)_n}{(\alpha+1)_n(\beta+\delta+1)_n(\gamma+1)_n}, :(\cdot)_n is the Pochhammer symbol.

Rodrigues-type formula

:\omega(x;\alpha,\beta,\gamma,\delta)\operatorname{R}_n(\lambda(x);\alpha,\beta,\gamma,\delta)=(\gamma+\delta+1)_n\frac{\nabla^n}{\nabla\lambda(x)^n}\omega(x;\alpha+n,\beta+n,\gamma+n,\delta), :where \nabla is the backward difference operator, :\lambda(x)=x(x+\gamma+\delta+1).

Generating functions

There are three generating functions for x\in{0,1,2,...,N} :when \beta+\delta+1=-N\quador\quad\gamma+1=-N, :{}_2F_1(-x,-x+\alpha-\gamma-\delta;\alpha+1;t){}2F_1(x+\beta+\delta+1,x+\gamma+1;\beta+1;t) :\quad=\sum{n=0}^N\frac{(\beta+\delta+1)_n(\gamma+1)_n}{(\beta+1)_nn!}\operatorname{R}_n(\lambda(x);\alpha,\beta,\gamma,\delta)t^n, :when \alpha+1=-N\quador\quad\gamma+1=-N, :{}_2F_1(-x,-x+\beta-\gamma;\beta+\delta+1;t){}2F_1(x+\alpha+1,x+\gamma+1;\alpha-\delta+1;t) :\quad=\sum{n=0}^N\frac{(\alpha+1)_n(\gamma+1)_n}{(\alpha-\delta+1)_nn!}\operatorname{R}_n(\lambda(x);\alpha,\beta,\gamma,\delta)t^n, :when \alpha+1=-N\quador\quad\beta+\delta+1=-N, :{}_2F_1(-x,-x-\delta;\gamma+1;t){}2F_1(x+\alpha+1;x+\beta+\gamma+1;\alpha+\beta-\gamma+1;t) :\quad=\sum{n=0}^N\frac{(\alpha+1)_n(\beta+\delta+1)_n}{(\alpha+\beta-\gamma+1)_nn!}\operatorname{R}_n(\lambda(x);\alpha,\beta,\gamma,\delta)t^n.

Connection formula for Wilson polynomials

When \alpha=a+b-1,\beta=c+d-1,\gamma=a+d-1,\delta=a-d,x\rightarrow-a+ix, :\operatorname{R}_n(\lambda(-a+ix);a+b-1,c+d-1,a+d-1,a-d)=\frac{\operatorname{W}_n(x^2;a,b,c,d)}{(a+b)_n(a+c)_n(a+d)_n}, :where \operatorname{W} are Wilson polynomials.

q-analog

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

References

1. Wilson, J.. (1978). "Hypergeometric series recurrence relations and some new orthogonal functions". Univ. Wisconsin, Madison.
2. Koornwinder, Tom H.. "Wilson Class: Definitions".
3. (1998). "The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue".
4. (1979). "A set of orthogonal polynomials that generalize the Racah coefficients or 6-j symbols". SIAM Journal on Mathematical Analysis.