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<p>68.184.28.65: /* Definition */ Wording.</p> <hr /> <div>{{distinguish|Al-Salam–Carlitz polynomials}}<br /> In mathematics, the '''Al-Salam–Chihara polynomials''' ''Q''&lt;sub&gt;''n''&lt;/sub&gt;(''x'';''a'',''b'';''q'') are a family of basic hypergeometric [[orthogonal polynomials]] in the basic [[Askey scheme]], introduced by {{harvs|txt|last=Al-Salam|authorlink=Waleed Al-Salam|last2=Chihara|author2-link=Theodore Seio Chihara|year=1976}}. {{harvs|txt | last1=Koekoek | first1=Roelof | last2=Lesky | first2=Peter A. | last3=Swarttouw | first3=René F. | title=Hypergeometric orthogonal polynomials and their q-analogues | doi=10.1007/978-3-642-05014-5 | 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=14.8}} give a detailed list of the properties of Al-Salam–Chihara polynomials.<br /> <br /> ==Definition==<br /> <br /> The Al-Salam–Chihara polynomials are given in terms of [[basic hypergeometric function]]s and the [[Pochhammer symbol]] by <br /> :&lt;math&gt; Q_n(x;a,b;q) = \frac{(ab;q)_n}{a^n}{}_3\phi_2(q^{-n}, ae^{i\theta}, ae^{-i\theta}; ab,0; q,q)&lt;/math&gt;<br /> where ''x''&amp;nbsp;=&amp;nbsp;cos(θ).<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=Al-Salam | first1=W. A. | last2=Chihara | first2=Theodore Seio | title=Convolutions of orthonormal polynomials | doi=10.1137/0507003 | mr=0399537 | year=1976 | journal=SIAM Journal on Mathematical Analysis | issn=0036-1410 | volume=7 | issue=1 | pages=16–28}}<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=18.28iii|title=Al-Salam–Chihara polynomials|first=Tom H. |last=Koornwinder|first2=Roderick S. C.|last2= Wong|first3=Roelof |last3=Koekoek||first4=René F. |last4=Swarttouw}}<br /> <br /> {{DEFAULTSORT:Al-Salam-Chihara polynomials}}<br /> [[Category:Orthogonal polynomials]]<br /> [[Category:Q-analogs]]<br /> [[Category:Special hypergeometric functions]]</div>

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