3rd order Continuous q Laguerre polynomials
连续q拉盖尔多项式(Continuous q-Laguerre polynomials)是一个以基本超几何函数定义的正交多项式[1]。
![{\displaystyle P_{n}^{(\alpha )}(x|q)={\frac {(q^{\alpha }+1;q)_{n}}{(q;q)_{n}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/62352deb2a15a0e70f0e3aab82f1967c96e8c85d)
极限关系
Q梅西纳-帕拉泽克多项式→连续q拉盖尔多项式
![{\displaystyle P_{n}(cos(\theta +\phi );q^{\alpha /2+1/2}|q)=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ecfa4317c12c97fba64a6caaf75ebc21994b4234)
阿拉-萨拉姆-迟哈剌多项式→连续q拉盖尔多项式
令连续q拉盖尔多项式中
,q→1,即得拉盖尔多项式
- 验证
3阶连续q拉盖尔多项式:
3阶广义拉盖尔多项式:
两者显然相等。
图集
CONTINUOUS Q LAGUERRE ABS COMPLEX 3D MAPLE PLOT
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CONTINUOUS Q LAGUERRE IM COMPLEX 3D MAPLE PLOT
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CONTINUOUS Q LAGUERRE RE COMPLEX 3D MAPLE PLOT
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CONTINUOUS Q LAGUERRE ABS DENSITY MAPLE PLOT
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CONTINUOUS Q LAGUERRE IM DENSITY MAPLE PLOT
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CONTINUOUS Q LAGUERRE RE DENSITY MAPLE PLOT
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参考文献
- 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
- ^ Roelof Koekoek, Peter Lesky, Rene Swarttouw,Hypergeometric Orthogonal Polynomials and Their q-Analogues, p514, Springer