羅傑斯多項式

羅傑斯多項式又稱連續q超球面多項式是一個以超幾何函數定義的超幾何正交多項式^[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${\displaystyle \theta =arccos(x)}$

極限關係

Q梅西納-帕拉澤克多項式→連續q超球面多項式

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

圖集

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

參考文獻

• Askey, Richard; Ismail, Mourad E. H., A generalization of ultraspherical polynomials, Erdős, Paul (編), 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, （原始內容存檔於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