Racah polynomials

${\displaystyle \sum _{y=0}^{N}\operatorname {R} _{n}(x;\alpha ,\beta ,\gamma ,\delta )\operatorname {R} _{m}(x;\alpha ,\beta ,\gamma ,\delta ){\frac {\gamma +\delta +1+2y}{\gamma +\delta +1+y}}\omega _{y}=h_{n}\operatorname {\delta } _{n,m},}$[1]

when ${\displaystyle \alpha +1=-N}$,

where ${\displaystyle \operatorname {R} }$ is the Racah polynomial,

${\displaystyle x=y(y+\gamma +\delta +1),}$

${\displaystyle \operatorname {\delta } _{n,m}}$ is the Kronecker delta function and the weight functions are

${\displaystyle \omega _{y}={\frac {(\alpha +1)_{y}(\beta +\delta +1)_{y}(\gamma +1)_{y}(\gamma +\delta +2)_{y}}{(-\alpha +\gamma +\delta +1)_{y}(-\beta +\gamma +1)_{y}(\delta +1)_{y}y!}},}$

and

${\displaystyle h_{n}={\frac {(-\beta )_{N}(\gamma +\delta +1)_{N}}{(-\beta +\gamma +1)_{N}(\delta +1)_{N}}}{\frac {(n+\alpha +\beta +1)_{n}n!}{(\alpha +\beta +2)_{2n}}}{\frac {(\alpha +\delta -\gamma +1)_{n}(\alpha -\delta +1)_{n}(\beta +1)_{n}}{(\alpha +1)_{n}(\beta +\delta +1)_{n}(\gamma +1)_{n}}},}$

${\displaystyle (\cdot )_{n}}$ is the Pochhammer symbol.

${\displaystyle \omega (x;\alpha ,\beta ,\gamma ,\delta )\operatorname {R} _{n}(\lambda (x);\alpha ,\beta ,\gamma ,\delta )=(\gamma +\delta +1)_{n}{\frac {\nabla ^{n}}{\nabla \lambda (x)^{n}}}\omega (x;\alpha +n,\beta +n,\gamma +n,\delta ),}$[2]

where ${\displaystyle \nabla }$ is the backward difference operator,

${\displaystyle \lambda (x)=x(x+\gamma +\delta +1).}$

There are three generating functions for ${\displaystyle x\in \{0,1,2,...,N\}}$

when ${\displaystyle \beta +\delta +1=-N\quad }$or${\displaystyle \quad \gamma +1=-N,}$

${\displaystyle {}_{2}F_{1}(-x,-x+\alpha -\gamma -\delta$ ;\alpha +1;t){}_{2}F_{1}(x+\beta +\delta +1,x+\gamma +1;\beta +1;t)}

${\displaystyle \quad =\sum _{n=0}^{N}{\frac {(\beta +\delta +1)_{n}(\gamma +1)_{n}}{(\beta +1)_{n}n!}}\operatorname {R} _{n}(\lambda (x);\alpha ,\beta ,\gamma ,\delta )t^{n},}$

when ${\displaystyle \alpha +1=-N\quad }$or${\displaystyle \quad \gamma +1=-N,}$

${\displaystyle {}_{2}F_{1}(-x,-x+\beta -\gamma$ ;\beta +\delta +1;t){}_{2}F_{1}(x+\alpha +1,x+\gamma +1;\alpha -\delta +1;t)}

${\displaystyle \quad =\sum _{n=0}^{N}{\frac {(\alpha +1)_{n}(\gamma +1)_{n}}{(\alpha -\delta +1)_{n}n!}}\operatorname {R} _{n}(\lambda (x);\alpha ,\beta ,\gamma ,\delta )t^{n},}$

when ${\displaystyle \alpha +1=-N\quad }$or${\displaystyle \quad \beta +\delta +1=-N,}$

${\displaystyle {}_{2}F_{1}(-x,-x-\delta$ ;\gamma +1;t){}_{2}F_{1}(x+\alpha +1;x+\beta +\gamma +1;\alpha +\beta -\gamma +1;t)}

${\displaystyle \quad =\sum _{n=0}^{N}{\frac {(\alpha +1)_{n}(\beta +\delta +1)_{n}}{(\alpha +\beta -\gamma +1)_{n}n!}}\operatorname {R} _{n}(\lambda (x);\alpha ,\beta ,\gamma ,\delta )t^{n}.}$

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

${\displaystyle \operatorname {R} _{n}(\lambda (-a+ix);a+b-1,c+d-1,a+d-1,a-d)={\frac {\operatorname {W} _{n}(x^{2};a,b,c,d)}{(a+b)_{n}(a+c)_{n}(a+d)_{n}}},}$

where ${\displaystyle \operatorname {W} }$ are Wilson polynomials.

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

${\displaystyle p_{n}(q^{-x}+q^{x+1}cd;a,b,c,d;q)={}_{4}\phi _{3}\left[{\begin{matrix}q^{-n}&abq^{n+1}&q^{-x}&q^{x+1}cd\\aq&bdq&cq\\\end{matrix}};q;q\right].}$

They are sometimes given with changes of variables as

${\displaystyle W_{n}(x;a,b,c,N;q)={}_{4}\phi _{3}\left[{\begin{matrix}q^{-n}&abq^{n+1}&q^{-x}&cq^{x-n}\\aq&bcq&q^{-N}\\\end{matrix}};q;q\right].}$

• Askey, Richard; Wilson, James (1979), "A set of orthogonal polynomials that generalize the Racah coefficients or 6-j symbols" (PDF), SIAM Journal on Mathematical Analysis, 10 (5): 1008–1016, doi:10.1137/0510092, ISSN 0036-1410, MR 0541097, archived from the original on September 25, 2017
• Wilson, J. (1978), Hypergeometric series recurrence relations and some new orthogonal functions, Ph.D. thesis, Univ. Wisconsin, Madison