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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
![{\displaystyle p_{n}(x(x+\gamma +\delta +1))={}_{4}F_{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].}](http://178.128.105.246/host-https-wikimedia.org/api/rest_v1/media/math/render/svg/2f8669e91de2c035849fc87d8ce46aa12600695e)
[1]
- when
,
- where
is the Racah polynomial,
![{\displaystyle x=y(y+\gamma +\delta +1),}](http://178.128.105.246/host-https-wikimedia.org/api/rest_v1/media/math/render/svg/6625a5ace02129a8ee52f8d27349e5c7f6de552e)
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!}},}](http://178.128.105.246/host-https-wikimedia.org/api/rest_v1/media/math/render/svg/f0396e7fc008550c40a8b91abd9a87d4cd2c0bf2)
- 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}}},}](http://178.128.105.246/host-https-wikimedia.org/api/rest_v1/media/math/render/svg/085135a5bc4867285de07882971f0dbcf8ae5d6d)
is the Pochhammer symbol.
[2]
- where
is the backward difference operator,
![{\displaystyle \lambda (x)=x(x+\gamma +\delta +1).}](http://178.128.105.246/host-https-wikimedia.org/api/rest_v1/media/math/render/svg/68a3d46ceeb22274e1863bc7cd651d50594e6b79)
Generating functions
[edit]
There are three generating functions for
- when
or![{\displaystyle \quad \gamma +1=-N,}](http://178.128.105.246/host-https-wikimedia.org/api/rest_v1/media/math/render/svg/a514f4cd839e2fbf80048efdaa3a4e8cf36d19b2)
![{\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)}](http://178.128.105.246/host-https-wikimedia.org/api/rest_v1/media/math/render/svg/1e65769f81268b69fcd18be6ef6a0ff87ee7ed43)
![{\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},}](http://178.128.105.246/host-https-wikimedia.org/api/rest_v1/media/math/render/svg/d78f35e076f3e8920f97110b45c3b993177b050d)
- when
or![{\displaystyle \quad \gamma +1=-N,}](http://178.128.105.246/host-https-wikimedia.org/api/rest_v1/media/math/render/svg/a514f4cd839e2fbf80048efdaa3a4e8cf36d19b2)
![{\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)}](http://178.128.105.246/host-https-wikimedia.org/api/rest_v1/media/math/render/svg/6b5f099c05f6802b2d87dcea123a2df7a804705b)
![{\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},}](http://178.128.105.246/host-https-wikimedia.org/api/rest_v1/media/math/render/svg/18d8365238ad5635b69ac619e7dc7d2d73247320)
- when
or![{\displaystyle \quad \beta +\delta +1=-N,}](http://178.128.105.246/host-https-wikimedia.org/api/rest_v1/media/math/render/svg/d3ccc4bd84d4de9bde3a6408a889b0bc0ae4a9f1)
![{\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)}](http://178.128.105.246/host-https-wikimedia.org/api/rest_v1/media/math/render/svg/60cc67c6c7754d18756263f2c79e5550f6f9c88c)
![{\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}.}](http://178.128.105.246/host-https-wikimedia.org/api/rest_v1/media/math/render/svg/123c869a04c247c9eacd104cfde5949e275abf4c)
When
![{\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}}},}](http://178.128.105.246/host-https-wikimedia.org/api/rest_v1/media/math/render/svg/48637deea4e0cb6edc6c12bfd0e29cb4f1d101f3)
- where
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].}](http://178.128.105.246/host-https-wikimedia.org/api/rest_v1/media/math/render/svg/2dc584f57d22b464c0732a0e61d35db040eedb8a)
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].}](http://178.128.105.246/host-https-wikimedia.org/api/rest_v1/media/math/render/svg/db8e78dfc0de0af60835bc8e8bcf8eaa6ae338c8)
- ^ Koornwinder, Tom H.; Wong, Roderick S. C.; Koekoek, Roelof; Swarttouw, René F. (2010), "Wilson Class: Definitions", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248.
- ^ Koekoek, Roelof; Swarttouw, René F. (1998), The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue
- 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