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
;
β β -->
|
q
)
=
(
β β -->
;
q
)
n
(
q
;
q
)
n
e
i
n
θ θ -->
2
ϕ ϕ -->
1
(
q
− − -->
n
,
β β -->
;
β β -->
− − -->
1
q
1
− − -->
n
;
q
,
q
β β -->
− − -->
1
e
− − -->
2
i
θ θ -->
)
{\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
θ θ -->
=
a
r
c
c
o
s
(
x
)
{\displaystyle \theta =arccos(x)}
Q梅西纳-帕拉泽克多项式 →连续q超球面多项式
P
n
(
c
o
s
ϕ ϕ -->
;
β β -->
|
q
)
=
C
n
(
c
o
s
ϕ ϕ -->
)
;
β β -->
|
q
)
{\displaystyle 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 978-3-7643-1288-6MR 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 978-0-521-83357-8MR 2128719 doi:10.2277/0521833574 Macdonald, I. G. , Affine Hecke algebras and orthogonal polynomials, Cambridge Tracts in Mathematics 157 , Cambridge University Press , 2003, ISBN 978-0-521-82472-9MR 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 978-3-7643-1288-6MR 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 978-0-521-83357-8MR 2128719 doi:10.2277/0521833574 Macdonald, I. G. , Affine Hecke algebras and orthogonal polynomials, Cambridge Tracts in Mathematics 157 , Cambridge University Press , 2003, ISBN 978-0-521-82472-9MR 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
