罗杰斯多项式

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^[1]

${\displaystyle C_{n}(x;\beta |q)={\frac {(\beta ;q)_{n}}{(q;q)_{n}}}e^{in\theta }{}_{2}\phi _{1}(q^{-n},\beta ;\beta ^{-1}q^{1-n};q,q\beta ^{-1}e^{-2i\theta })}$

其中 x = cos(θ).即${\displaystyle \theta =arccos(x)}$

极限关系

Q梅西纳-帕拉泽克多项式→连续q超球面多项式

${\displaystyle P_{n}(cos\phi ;\beta |q)=C_{n}(cos\phi );\beta |q)}$ 

图集

ROGERS POLYNOMIALS ABS COMPLEX3D MAPLE PLOT

ROGERS POLYNOMIALS IM COMPLEX3D MAPLE PLOT

ROGERS POLYNOMIALS RE COMPLEX3D MAPLE PLOT

ROGERS POLYNOMIALS ABS DENSITY MAPLE PLOT

ROGERS POLYNOMIALS IM DENSITY MAPLE PLOT

ROGERS POLYNOMIALS RE DENSITY MAPLE PLOT

参考文献

• Askey, Richard; Ismail, Mourad E. H., A generalization of ultraspherical polynomials, Erdős, Paul (编), , Basel, Boston, Berlin: Birkhäuser: 55–78, 1983 [2015-01-30], ISBN 978-3-7643-1288-6, MR 0820210, （原始内容存档于2014-01-05） 
• Gasper, George; Rahman, Mizan, Basic hypergeometric series, Encyclopedia of Mathematics and its Applications 96 2nd, Cambridge University Press, 2004, ISBN 978-0-521-83357-8, MR 2128719, doi:10.2277/0521833574 
• Macdonald, I. G., Affine Hecke algebras and orthogonal polynomials, Cambridge Tracts in Mathematics 157, Cambridge University Press, 2003, ISBN 978-0-521-82472-9, MR 1976581, doi:10.1017/CBO9780511542824 
• Rogers, L. J., On the expansion of some infinite products, Proc. London Math. Soc., 1892, 24 (1): 337–352, JFM 25.0432.01, doi:10.1112/plms/s1-24.1.337 
• Rogers, L. J., Second Memoir on the Expansion of certain Infinite Products, Proc. London Math. Soc., 1893, 25 (1): 318–343, doi:10.1112/plms/s1-25.1.318 
• Rogers, L. J., Third Memoir on the Expansion of certain Infinite Products, Proc. London Math. Soc., 1894, 26 (1): 15–32, doi:10.1112/plms/s1-26.1.15 

参考文献

1. Roelof Koekoek, Hypergeometric Orthogonal Polynomials and its q-Analogues,p469,Springer 2010

• Askey, Richard; Ismail, Mourad E. H., A generalization of ultraspherical polynomials, Erdős, Paul (编), , Basel, Boston, Berlin: Birkhäuser: 55–78, 1983 [2015-01-30], ISBN 978-3-7643-1288-6, MR 0820210, （原始内容存档于2014-01-05） 
• Gasper, George; Rahman, Mizan, Basic hypergeometric series, Encyclopedia of Mathematics and its Applications 96 2nd, Cambridge University Press, 2004, ISBN 978-0-521-83357-8, MR 2128719, doi:10.2277/0521833574 
• Macdonald, I. G., Affine Hecke algebras and orthogonal polynomials, Cambridge Tracts in Mathematics 157, Cambridge University Press, 2003, ISBN 978-0-521-82472-9, MR 1976581, doi:10.1017/CBO9780511542824 
• Rogers, L. J., On the expansion of some infinite products, Proc. London Math. Soc., 1892, 24 (1): 337–352, JFM 25.0432.01, doi:10.1112/plms/s1-24.1.337 
• Rogers, L. J., Second Memoir on the Expansion of certain Infinite Products, Proc. London Math. Soc., 1893, 25 (1): 318–343, doi:10.1112/plms/s1-25.1.318 
• Rogers, L. J., Third Memoir on the Expansion of certain Infinite Products, Proc. London Math. Soc., 1894, 26 (1): 15–32, doi:10.1112/plms/s1-26.1.15