查看“︁罗杰斯多项式”︁的源代码

'''罗杰斯多项式'''又称[[连续q超球面多项式]]是一个以[[超几何函数]]定义的超几何[[正交多项式]]<ref name=R>Roelof Koekoek, Hypergeometric Orthogonal Polynomials and its q-Analogues,p469,Springer 2010</ref>

:<math> C_n(x;\beta|q) = \frac{(\beta;q)_n}{(q;q)_n}e^{in\theta} {}_2\phi_1(q^{-n},\beta;\beta^{-1}q^{1-n};q,q\beta^{-1}e^{-2i\theta})</math>

其中 ''x''&nbsp;=&nbsp;cos(''θ'').即<math>\theta= arccos(x)</math>
==极限关系==
[[Q梅西纳-帕拉泽克多项式]]→[[连续q超球面多项式]]

<math>P_{n}(cos\phi;\beta|q)=C_{n}(cos\phi);\beta|q)</math>

==图集==
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|[[File:ROGERS POLYNOMIALS ABS COMPLEX3D MAPLE PLOT.gif|thumb|ROGERS POLYNOMIALS ABS COMPLEX3D MAPLE PLOT]]
|[[File:ROGERS POLYNOMIALS IM COMPLEX3D MAPLE PLOT.gif|thumb|ROGERS POLYNOMIALS IM COMPLEX3D MAPLE PLOT]]
|[[File:ROGERS POLYNOMIALS RE COMPLEX3D MAPLE PLOT.gif|thumb|ROGERS POLYNOMIALS RE COMPLEX3D MAPLE PLOT]]
|}
{|
|[[File:ROGERS POLYNOMIALS ABS DENSITY MAPLE PLOT.gif|thumb|ROGERS POLYNOMIALS ABS DENSITY MAPLE PLOT]]
|[[File:ROGERS POLYNOMIALS IM DENSITY MAPLE PLOT.gif|thumb|ROGERS POLYNOMIALS IM DENSITY MAPLE PLOT]]
|[[File:ROGERS POLYNOMIALS RE DENSITY MAPLE PLOT.gif|thumb|ROGERS POLYNOMIALS RE DENSITY MAPLE PLOT]]
|}



==参考文献==

*{{Citation | last1=Askey | first1=Richard | last2=Ismail | first2=Mourad E. H. | editor1-last=Erdős | editor1-first=Paul | title=Studies in pure mathematics. To the memory of Paul Turán. | url=http://books.google.com/books?id=WePuAAAAMAAJ | publisher=Birkhäuser | location=Basel, Boston, Berlin | isbn=978-3-7643-1288-6 | mr=820210 | year=1983 | chapter=A generalization of ultraspherical polynomials | pages=55–78 | accessdate=2015-01-30 | archive-date=2014-01-05 | archive-url=https://web.archive.org/web/20140105051653/http://books.google.com/books?id=WePuAAAAMAAJ }}
*{{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}}
*{{Citation | last1=Macdonald | first1=I. G. | author1-link=Ian G. Macdonald | title=Affine Hecke algebras and orthogonal polynomials | publisher=[[Cambridge University Press]] | series=Cambridge Tracts in Mathematics | isbn=978-0-521-82472-9 | doi=10.1017/CBO9780511542824 | mr=1976581 | year=2003 | volume=157}}
*{{Citation | last1=Rogers | first1=L. J. | title=On the expansion of some infinite products | doi=10.1112/plms/s1-24.1.337  | jfm=25.0432.01 | year=1892 | journal=Proc. London Math. Soc. | volume=24 | issue=1 | pages=337–352 }}
*{{Citation | last1=Rogers | first1=L. J. | title=Second Memoir on the Expansion of certain Infinite Products  | doi=10.1112/plms/s1-25.1.318 | year=1893 | journal=Proc. London Math. Soc. | volume=25 | issue=1 | pages=318–343}}
*{{Citation | last1=Rogers | first1=L. J. | title=Third Memoir on the Expansion of certain Infinite Products  | doi=10.1112/plms/s1-26.1.15  | year=1894 | journal=Proc. London Math. Soc. | volume=26 | issue=1 | pages=15–32}}

[[Category:正交多项式]]
[[Category:Q-模拟]]

==参考文献==
<references/>
*{{Citation | last1=Askey | first1=Richard | last2=Ismail | first2=Mourad E. H. | editor1-last=Erdős | editor1-first=Paul | title=Studies in pure mathematics. To the memory of Paul Turán. | url=http://books.google.com/books?id=WePuAAAAMAAJ | publisher=Birkhäuser | location=Basel, Boston, Berlin | isbn=978-3-7643-1288-6 | mr=820210 | year=1983 | chapter=A generalization of ultraspherical polynomials | pages=55–78 | accessdate=2015-01-30 | archive-date=2014-01-05 | archive-url=https://web.archive.org/web/20140105051653/http://books.google.com/books?id=WePuAAAAMAAJ }}
*{{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}}
*{{Citation | last1=Macdonald | first1=I. G. | author1-link=Ian G. Macdonald | title=Affine Hecke algebras and orthogonal polynomials | publisher=[[Cambridge University Press]] | series=Cambridge Tracts in Mathematics | isbn=978-0-521-82472-9 | doi=10.1017/CBO9780511542824 | mr=1976581 | year=2003 | volume=157}}
*{{Citation | last1=Rogers | first1=L. J. | title=On the expansion of some infinite products | doi=10.1112/plms/s1-24.1.337  | jfm=25.0432.01 | year=1892 | journal=Proc. London Math. Soc. | volume=24 | issue=1 | pages=337–352 }}
*{{Citation | last1=Rogers | first1=L. J. | title=Second Memoir on the Expansion of certain Infinite Products  | doi=10.1112/plms/s1-25.1.318 | year=1893 | journal=Proc. London Math. Soc. | volume=25 | issue=1 | pages=318–343}}
*{{Citation | last1=Rogers | first1=L. J. | title=Third Memoir on the Expansion of certain Infinite Products  | doi=10.1112/plms/s1-26.1.15  | year=1894 | journal=Proc. London Math. Soc. | volume=26 | issue=1 | pages=15–32}}

[[Category:正交多项式]]
[[Category:Q-模拟]]