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]

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 $\theta =arccos(x)$

极限关系

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

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

Askey, Richard; Ismail, Mourad E. H., A generalization of ultraspherical polynomials, Erdős, Paul (编), Studies in pure mathematics. To the memory of Paul Turán., 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

^ 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), Studies in pure mathematics. To the memory of Paul Turán., 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