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Askey–Wilson polynomials
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<h2 id="proof">Proof</h2> <p>This result can be proven since it is known that </p> p n ( cos θ ) = p n ( cos θ ; a , b , c , d ∣ q ) {\displaystyle p_{n}(\cos {\theta })=p_{n}(\cos {\theta };a,b,c,d\mid q)} <p>and using the definition of the <i>q</i>-Pochhammer symbol </p> p n ( cos θ ) = a − n ∑ ℓ = 0 n q ℓ ( a b q ℓ , a c q ℓ , a d q ℓ ; q ) n − ℓ × ( q − n , a b c d q n − 1 ; q ) ℓ ( q ; q ) ℓ ∏ j = 0 ℓ − 1 ( 1 − 2 a q j cos θ + a 2 q 2 j ) {\displaystyle p_{n}(\cos {\theta })=a^{-n}\sum _{\ell =0}^{n}q^{\ell }\left(abq^{\ell },acq^{\ell },adq^{\ell };q\right)_{n-\ell }\times {\frac {\left(q^{-n},abcdq^{n-1};q\right)_{\ell }}{(q;q)_{\ell }}}\prod _{j=0}^{\ell -1}\left(1-2aq^{j}\cos {\theta }+a^{2}q^{2j}\right)} <p>which leads to the conclusion that it equals </p> a − n ( a b , a c , a d ; q ) n 4 ϕ 3 [ q − n a b c d q n − 1 a e i θ a e − i θ a b a c a d ; q , q ] {\displaystyle a^{-n}(ab,ac,ad;q)_{n}\;_{4}\phi _{3}\left[{\begin{matrix}q^{-n}&abcdq^{n-1}&ae^{i\theta }&ae^{-i\theta }\\ab&ac&ad\end{matrix}};q,q\right]} <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Askey_scheme/cmELnDGN">Askey scheme</a></li></ul> <ul><li><a href="/facts/Richard_Askey/r6ScCdJa">Askey, Richard</a>; <a href="/facts/James_A._Wilson/xxjC8beA">Wilson, James</a> (1985), <a href="https://books.google.com/books?id=9q9o03nD_xsC">"Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials"</a>, <i>Memoirs of the American Mathematical Society</i>, 54 (319): iv+55, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1090%2Fmemo%2F0319">10.1090/memo/0319</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-8218-2321-7, <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0065-9266">0065-9266</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0783216">0783216</a></li> <li>Gasper, George; Rahman, Mizan (2004), <i>Basic hypergeometric series</i>, Encyclopedia of Mathematics and its Applications, vol. 96 (2nd ed.), <a href="/facts/Cambridge_University_Press/fgEBSSRq">Cambridge University Press</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-521-83357-8, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2128719">2128719</a></li> <li>Koornwinder, Tom H.; Wong, Roderick S. C.; Koekoek, Roelof; Swarttouw, René F. (2010), <a href="http://dlmf.nist.gov/18.28">"Askey-Wilson class"</a>, in <a href="/facts/Frank_W._J._Olver/nQYQVRHc">Olver, Frank W. J.</a>; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), <i><a href="/facts/Digital_Library_of_Mathematical_Functions/ymNQBbgK">NIST Handbook of Mathematical Functions</a></i>, Cambridge University Press, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-521-19225-5, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2723248">2723248</a>.</li> <li>Koornwinder, Tom H. (2012), "Askey-Wilson polynomial", <i>Scholarpedia</i>, 7 (7): 7761, <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/2012SchpJ...7.7761K">2012SchpJ...7.7761K</a>, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.4249%2Fscholarpedia.7761">10.4249/scholarpedia.7761</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Askey & Wilson (1985). - Askey, Richard; Wilson, James (1985), "Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials", Memoirs of the American Mathematical Society, 54 (319): iv+55, doi:10.1090/memo/0319, ISBN 978-0-8218-2321-7, ISSN 0065-9266, MR 0783216 <a href="https://books.google.com/books?id=9q9o03nD_xsC" target="_blank">https://books.google.com/books?id=9q9o03nD_xsC</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> </ol>