https://en.formulasearchengine.com/api.php?action=feedcontributions&feedformat=atom&user=153.90.118.189 formulasearchengine - User contributions [en] 2026-05-31T15:19:47Z User contributions MediaWiki 1.43.0-wmf.28 https://en.formulasearchengine.com/index.php?title=Klinkenberg_correction&diff=22971 Klinkenberg correction 2013-08-29T22:44:06Z <p>153.90.118.189: </p> <hr /> <div>{{DISPLAYTITLE: ''q''-Racah polynomials}}<br /> In mathematics, the '''''q''-Racah polynomials''' are a family of basic hypergeometric [[orthogonal polynomials]] in the basic [[Askey scheme]], introduced by {{harvtxt|Askey|Wilson|1979}}. {{harvs|txt | last1=Koekoek | first1=Roelof | last2=Lesky | first2=Peter A. | last3=Swarttouw | first3=René F. | title=Hypergeometric orthogonal polynomials and their q-analogues | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Springer Monographs in Mathematics | isbn=978-3-642-05013-8 | doi=10.1007/978-3-642-05014-5 | mr=2656096 | year=2010|loc=14}} give a detailed list of their properties.<br /> <br /> ==Definition==<br /> <br /> The polynomials are given in terms of [[basic hypergeometric function]]s and the [[Pochhammer symbol]] by <br /> :&lt;math&gt;p_n(q^{-x}+q^{x+1}cd;a,b,c,d;q) = {}_4\phi_3\left[\begin{matrix} q^{-n} &amp;abq^{n+1}&amp;q^{-x}&amp;q^{x+1}cd\\<br /> aq&amp;bdq&amp;cq\\ \end{matrix};q;q\right]&lt;/math&gt;<br /> They are sometimes given with changes of variables as<br /> :&lt;math&gt;W_n(x;a,b,c,N;q) = {}_4\phi_3\left[\begin{matrix} q^{-n} &amp;abq^{n+1}&amp;q^{-x}&amp;cq^{x-n}\\<br /> aq&amp;bcq&amp;q^{-N}\\ \end{matrix};q;q\right]&lt;/math&gt;<br /> <br /> ==Orthogonality==<br /> {{Empty section|date=September 2011}}<br /> <br /> ==Recurrence and difference relations==<br /> {{Empty section|date=September 2011}}<br /> <br /> ==Rodrigues formula==<br /> {{Empty section|date=September 2011}}<br /> <br /> ==Generating function==<br /> {{Empty section|date=September 2011}}<br /> <br /> ==Relation to other polynomials==<br /> {{Empty section|date=September 2011}}<br /> <br /> ==References==<br /> <br /> *{{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}}<br /> *{{Citation | last1=Gasper | first1=George | last2=Rahman | first2=Mizan | title=Basic hypergeometric series | publisher=[[Cambridge University Press]] | edition=2nd | series=Encyclopedia of Mathematics and its Applications | isbn=978-0-521-83357-8 | doi=10.2277/0521833574 | mr=2128719 | year=2004 | volume=96}}<br /> *{{Citation | last1=Koekoek | first1=Roelof | last2=Lesky | first2=Peter A. | last3=Swarttouw | first3=René F. | title=Hypergeometric orthogonal polynomials and their q-analogues | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Springer Monographs in Mathematics | isbn=978-3-642-05013-8 | doi=10.1007/978-3-642-05014-5 | mr=2656096 | year=2010}}<br /> *{{dlmf|id=18|title=|first=Tom H. |last=Koornwinder|first2=Roderick S. C.|last2= Wong|first3=Roelof |last3=Koekoek||first4=René F. |last4=Swarttouw}}<br /> <br /> [[Category:Orthogonal polynomials]]<br /> [[Category:Q-analogs]]<br /> [[Category:Special hypergeometric functions]]</div> 153.90.118.189