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

[math]\displaystyle{ 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]. }[/math]

Orthogonality

[math]\displaystyle{ \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}, }[/math]

when [math]\displaystyle{ \alpha+1=-N }[/math],

where [math]\displaystyle{ \operatorname{R} }[/math] is the Racah polynomial,

[math]\displaystyle{ x=y(y+\gamma+\delta+1), }[/math]

[math]\displaystyle{ \operatorname{\delta}_{n,m} }[/math] is the Kronecker delta function and the weight functions are

[math]\displaystyle{ \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!}, }[/math]

and

[math]\displaystyle{ 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}, }[/math]

[math]\displaystyle{ (\cdot)_n }[/math] is the Pochhammer symbol.

Rodrigues-type formula

[math]\displaystyle{ \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), }[/math]

where [math]\displaystyle{ \nabla }[/math] is the backward difference operator,

[math]\displaystyle{ \lambda(x)=x(x+\gamma+\delta+1). }[/math]

Generating functions

There are three generating functions for [math]\displaystyle{ x\in\{0,1,2,...,N\} }[/math]

when [math]\displaystyle{ \beta+\delta+1=-N\quad }[/math]or[math]\displaystyle{ \quad\gamma+1=-N, }[/math]

[math]\displaystyle{ {}_2F_1(-x,-x+\alpha-\gamma-\delta;\alpha+1;t){}_2F_1(x+\beta+\delta+1,x+\gamma+1;\beta+1;t) }[/math]

[math]\displaystyle{ \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, }[/math]

when [math]\displaystyle{ \alpha+1=-N\quad }[/math]or[math]\displaystyle{ \quad\gamma+1=-N, }[/math]

[math]\displaystyle{ {}_2F_1(-x,-x+\beta-\gamma;\beta+\delta+1;t){}_2F_1(x+\alpha+1,x+\gamma+1;\alpha-\delta+1;t) }[/math]

[math]\displaystyle{ \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, }[/math]

when [math]\displaystyle{ \alpha+1=-N\quad }[/math]or[math]\displaystyle{ \quad\beta+\delta+1=-N, }[/math]

[math]\displaystyle{ {}_2F_1(-x,-x-\delta;\gamma+1;t){}_2F_1(x+\alpha+1;x+\beta+\gamma+1;\alpha+\beta-\gamma+1;t) }[/math]

[math]\displaystyle{ \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. }[/math]

Connection formula for Wilson polynomials

When [math]\displaystyle{ \alpha=a+b-1,\beta=c+d-1,\gamma=a+d-1,\delta=a-d,x\rightarrow-a+ix, }[/math]

[math]\displaystyle{ \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}, }[/math]

where [math]\displaystyle{ \operatorname{W} }[/math] are Wilson polynomials.

q-analog

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

[math]\displaystyle{ 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]. }[/math]

They are sometimes given with changes of variables as

[math]\displaystyle{ 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]. }[/math]

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, https://apps.dtic.mil/sti/pdfs/ADA054552.pdf 
• Wilson, J. (1978), Hypergeometric series recurrence relations and some new orthogonal functions, Ph.D. thesis, Univ. Wisconsin, Madison