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Racah polynomials
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<h2 id="orthogonality">Orthogonality</h2> ∑ y = 0 N R n ( x ; α , β , γ , δ ) R m ( x ; α , β , γ , δ ) γ + δ + 1 + 2 y γ + δ + 1 + y ω y = h n δ n , m , {\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},} <a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a> when α + 1 = − N {\displaystyle \alpha +1=-N} , where R {\displaystyle \operatorname {R} } is the Racah polynomial, x = y ( y + γ + δ + 1 ) , {\displaystyle x=y(y+\gamma +\delta +1),} δ n , m {\displaystyle \operatorname {\delta } _{n,m}} is the <a href="/facts/Kronecker_delta_function/gcGAy0bm">Kronecker delta function</a> and the weight functions are ω y = ( α + 1 ) y ( β + δ + 1 ) y ( γ + 1 ) y ( γ + δ + 2 ) y ( − α + γ + δ + 1 ) y ( − β + γ + 1 ) y ( δ + 1 ) y y ! , {\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 h n = ( − β ) N ( γ + δ + 1 ) N ( − β + γ + 1 ) N ( δ + 1 ) N ( n + α + β + 1 ) n n ! ( α + β + 2 ) 2 n ( α + δ − γ + 1 ) n ( α − δ + 1 ) n ( β + 1 ) n ( α + 1 ) n ( β + δ + 1 ) n ( γ + 1 ) n , {\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}}},} ( ⋅ ) n {\displaystyle (\cdot )_{n}} is the <a href="/facts/Pochhammer_symbol/sAxnmGoC">Pochhammer symbol</a>. <h2 id="rodrigues-type-formula">Rodrigues-type formula</h2> ω ( x ; α , β , γ , δ ) R n ( λ ( x ) ; α , β , γ , δ ) = ( γ + δ + 1 ) n ∇ n ∇ λ ( x ) n ω ( x ; α + n , β + n , γ + n , δ ) , {\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 ),} <a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> where ∇ {\displaystyle \nabla } is the <a href="/facts/Backward_difference_operator/aY1txFhj">backward difference operator</a>, λ ( x ) = x ( x + γ + δ + 1 ) . {\displaystyle \lambda (x)=x(x+\gamma +\delta +1).} <h2 id="generating-functions">Generating functions</h2> <p>There are three generating functions for x ∈ { 0 , 1 , 2 , . . . , N } {\displaystyle x\in \{0,1,2,...,N\}} </p> when β + δ + 1 = − N {\displaystyle \beta +\delta +1=-N\quad } or γ + 1 = − N , {\displaystyle \quad \gamma +1=-N,} 2 F 1 ( − x , − x + α − γ − δ ; α + 1 ; t ) 2 F 1 ( x + β + δ + 1 , x + γ + 1 ; β + 1 ; t ) {\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)} = ∑ n = 0 N ( β + δ + 1 ) n ( γ + 1 ) n ( β + 1 ) n n ! R n ( λ ( x ) ; α , β , γ , δ ) t n , {\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 α + 1 = − N {\displaystyle \alpha +1=-N\quad } or γ + 1 = − N , {\displaystyle \quad \gamma +1=-N,} 2 F 1 ( − x , − x + β − γ ; β + δ + 1 ; t ) 2 F 1 ( x + α + 1 , x + γ + 1 ; α − δ + 1 ; t ) {\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)} = ∑ n = 0 N ( α + 1 ) n ( γ + 1 ) n ( α − δ + 1 ) n n ! R n ( λ ( x ) ; α , β , γ , δ ) t n , {\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 α + 1 = − N {\displaystyle \alpha +1=-N\quad } or β + δ + 1 = − N , {\displaystyle \quad \beta +\delta +1=-N,} 2 F 1 ( − x , − x − δ ; γ + 1 ; t ) 2 F 1 ( x + α + 1 ; x + β + γ + 1 ; α + β − γ + 1 ; t ) {\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)} = ∑ n = 0 N ( α + 1 ) n ( β + δ + 1 ) n ( α + β − γ + 1 ) n n ! R n ( λ ( x ) ; α , β , γ , δ ) t n . {\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}.} <h2 id="connection-formula-for-wilson-polynomials">Connection formula for Wilson polynomials</h2> <p>When α = a + b − 1 , β = c + d − 1 , γ = a + d − 1 , δ = a − d , x → − a + i x , {\displaystyle \alpha =a+b-1,\beta =c+d-1,\gamma =a+d-1,\delta =a-d,x\rightarrow -a+ix,} </p> R n ( λ ( − a + i x ) ; a + b − 1 , c + d − 1 , a + d − 1 , a − d ) = W n ( x 2 ; a , b , c , d ) ( a + b ) n ( a + c ) n ( a + d ) n , {\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 W {\displaystyle \operatorname {W} } are Wilson polynomials. <h2 id="q-analog">q-analog</h2> <p>Askey & Wilson (1979) introduced the <i>q</i>-Racah polynomials defined in terms of <a href="/facts/Basic_hypergeometric_function/LuW9vAIs">basic hypergeometric functions</a> by </p> p n ( q − x + q x + 1 c d ; a , b , c , d ; q ) = 4 ϕ 3 [ q − n a b q n + 1 q − x q x + 1 c d a q b d q c q ; q ; q ] . {\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].} <p>They are sometimes given with changes of variables as </p> W n ( x ; a , b , c , N ; q ) = 4 ϕ 3 [ q − n a b q n + 1 q − x c q x − n a q b c q q − N ; q ; q ] . {\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].} <ul><li>Askey, Richard; Wilson, James (1979), <a href="https://apps.dtic.mil/sti/pdfs/ADA054552.pdf">"A set of orthogonal polynomials that generalize the Racah coefficients or 6-j symbols"</a> (PDF), <i>SIAM Journal on Mathematical Analysis</i>, 10 (5): 1008–1016, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1137%2F0510092">10.1137/0510092</a>, <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0036-1410">0036-1410</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0541097">0541097</a>, <a href="https://web.archive.org/web/20170925235936/http://www.dtic.mil/get-tr-doc/pdf?AD=ADA054552">archived</a> from the original on September 25, 2017</li> <li>Wilson, J. (1978), <i>Hypergeometric series recurrence relations and some new orthogonal functions</i>, Ph.D. thesis, Univ. Wisconsin, Madison</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>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. <a href="978-0-521-19225-5" target="_blank">978-0-521-19225-5</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Koekoek, Roelof; Swarttouw, René F. (1998), The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue <a href="https://fa.ewi.tudelft.nl/~koekoek/askey/ch1/par2/par2.html" target="_blank">https://fa.ewi.tudelft.nl/~koekoek/askey/ch1/par2/par2.html</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> </ol>