连续q雅可比多项式

连续q雅可比多项式定义如下

$P_{n}^{(\alpha ;\beta )}(x|q)=$${\frac {(q^{\alpha +1};q)_{n}}{(q;q;)_{n}}}$

• $_{4}\Phi _{3}$$(q^{-n},q^{n+\alpha +\beta +1};q^{\alpha /2+1/4}e^{i\theta };q^{\alpha /2+1/4}e^{-i\theta })$

极限关系

连续q雅可比多项式→连续q拉盖尔多项式

$\lim _{\beta \to \infty }P_{n}^{(\alpha ;\beta )}(x|q)=P_{n}^{(\alpha )}(x|q)$

连续q雅可比多项式→雅科比多项式

$\lim _{q\to 1}P_{n}^{(\alpha ;\beta )}(x|q)=P_{n}^{(\alpha ;\beta )}(x|q)$

图集

CONTINUOUS Q-JACOBI ABS COMPLEX 3D MAPLE PLOT

CONTINUOUS Q-JACOBI IM COMPLEX 3D MAPLE PLOT

CONTINUOUS Q-JACOBI RE COMPLEX 3D MAPLE PLOT

CONTINUOUS Q-JACOBI ABS DENSITY MAPLE PLOT

CONTINUOUS Q-JACOBI IM DENSITY MAPLE PLOT

CONTINUOUS Q-JACOBI RE DENSITY MAPLE PLOT

参考文献

• Askey, Richard; Wilson, James, Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials, Memoirs of the American Mathematical Society, 1985, 54 (319): iv+55, ISBN 978-0-8218-2321-7, ISSN 0065-9266, MR 0783216, doi:10.1090/memo/0319 
• 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 
• Koekoek, Roelof; Lesky, Peter A.; Swarttouw, René F., Hypergeometric orthogonal polynomials and their q-analogues, Springer Monographs in Mathematics, Berlin, New York: Springer-Verlag, 2010, ISBN 978-3-642-05013-8, MR 2656096, doi:10.1007/978-3-642-05014-5 
• Koornwinder, Tom H.; Wong, Roderick S. C.; Koekoek, Roelof; Swarttouw, René F., http://dlmf.nist.gov/18, Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (编), NIST Handbook of Mathematical Functions, Cambridge University Press, 2010, ISBN 978-0521192255, MR2723248 
• Rahman, Mizan, The linearization of the product of continuous q-Jacobi polynomials, Canadian Journal of Mathematics, 1981, 33 (4): 961–987, ISSN 0008-414X, MR 0634153, doi:10.4153/CJM-1981-076-8