Racah polynomials

:$\backslash sum\_\{y=0\}^N\backslash operatorname\{R\}\_n(x;\backslash alpha,\backslash beta,\backslash gamma,\backslash delta)$

\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},{{dlmf|id=18.25#iii|title=Wilson Class: Definitions|first=Tom H. |last=Koornwinder|first2=Roderick S. C.|last2= Wong|first3=Roelof |last3=Koekoek||first4=René F. |last4=Swarttouw}}

:when $\backslash alpha+1=-N$,

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

:$x=y(y+\backslash gamma+\backslash delta+1),$

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

:$\backslash omega\_y=\backslash frac\{(\backslash alpha+1)\_y(\backslash beta+\backslash delta+1)\_y(\backslash gamma+1)\_y(\backslash gamma+\backslash delta+2)\_y\}\{(-\backslash alpha+\backslash gamma+\backslash delta+1)\_y(-\backslash beta+\backslash gamma+1)\_y(\backslash delta+1)\_yy!\},$

:and

:$h\_n=\backslash frac\{(-\backslash beta)\_N(\backslash gamma+\backslash delta+1)\_N\}\{(-\backslash beta+\backslash gamma+1)\_N(\backslash delta+1)\_N\}\backslash frac\{(n+\backslash alpha+\backslash beta+1)\_nn!\}\{(\backslash alpha+\backslash beta+2)\_\{2n\}\}\backslash frac\{(\backslash alpha+\backslash delta-\backslash gamma+1)\_n(\backslash alpha-\backslash delta+1)\_n(\backslash beta+1)\_n\}\{(\backslash alpha+1)\_n(\backslash beta+\backslash delta+1)\_n(\backslash gamma+1)\_n\},$

:$(\backslash cdot)\_n$ is the Pochhammer symbol.

:$\backslash omega(x;\backslash alpha,\backslash beta,\backslash gamma,\backslash delta)\backslash operatorname\{R\}\_n(\backslash lambda(x);\backslash alpha,\backslash beta,\backslash gamma,\backslash delta)=(\backslash gamma+\backslash delta+1)\_n\backslash frac\{\backslash nabla^n\}\{\backslash nabla\backslash lambda(x)^n\}\backslash omega(x;\backslash alpha+n,\backslash beta+n,\backslash gamma+n,\backslash delta),${{Citation | last1=Koekoek | first1=Roelof | last2=Swarttouw | first2=René F. | title=The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue | year=1998 | url=https://fa.ewi.tudelft.nl/~koekoek/askey/ch1/par2/par2.html}}

:where $\backslash nabla$ is the backward difference operator,

:$\backslash lambda(x)=x(x+\backslash gamma+\backslash delta+1).$

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

:when $\backslash beta+\backslash delta+1=-N\backslash quad$or$\backslash quad\backslash gamma+1=-N,$

:$\{\}\_2F\_1(-x,-x+\backslash alpha-\backslash gamma-\backslash delta;\backslash alpha+1;t)\{\}\_2F\_1(x+\backslash beta+\backslash delta+1,x+\backslash gamma+1;\backslash beta+1;t)$

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

:when $\backslash alpha+1=-N\backslash quad$or$\backslash quad\backslash gamma+1=-N,$

:$\{\}\_2F\_1(-x,-x+\backslash beta-\backslash gamma;\backslash beta+\backslash delta+1;t)\{\}\_2F\_1(x+\backslash alpha+1,x+\backslash gamma+1;\backslash alpha-\backslash delta+1;t)$

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

:when $\backslash alpha+1=-N\backslash quad$or$\backslash quad\backslash beta+\backslash delta+1=-N,$

:$\{\}\_2F\_1(-x,-x-\backslash delta;\backslash gamma+1;t)\{\}\_2F\_1(x+\backslash alpha+1;x+\backslash beta+\backslash gamma+1;\backslash alpha+\backslash beta-\backslash gamma+1;t)$

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

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

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

:where $\backslash operatorname\{W\}$ are Wilson polynomials.

{{harvtxt|Askey|Wilson|1979}} introduced the q-Racah polynomials defined in terms of basic hypergeometric functions by

:

aq&bdq&cq\\ \end{matrix};q;q\right].

They are sometimes given with changes of variables as

:

aq&bcq&q^{-N}\\ \end{matrix};q;q\right].

{{Reflist}}

• {{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 |mr=541097 | year=1979 | journal=SIAM Journal on Mathematical Analysis | issn=0036-1410 | volume=10 | issue=5 | pages=1008–1016| url=https://apps.dtic.mil/sti/pdfs/ADA054552.pdf | archive-url=https://web.archive.org/web/20170925235936/http://www.dtic.mil/get-tr-doc/pdf?AD=ADA054552 | url-status=live | archive-date=September 25, 2017 }}
• {{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}}

Category:Orthogonal polynomials

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