连续q拉盖尔多项式

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]。

3rd order Continuous q Laguerre polynomials

${\displaystyle P_{n}^{(\alpha )}(x|q)={\frac {(q^{\alpha }+1;q)_{n}}{(q;q)_{n}}}}$${\displaystyle _{3}\Phi _{2}(q^{-n},q^{\alpha /2+1/4}e^{i\theta },q^{\alpha /2+1/4}*e^{-i\theta };q^{\alpha +1},0|q,q)}$

极限关系

Q梅西纳-帕拉泽克多项式→连续q拉盖尔多项式

${\displaystyle P_{n}(cos(\theta +\phi );q^{\alpha /2+1/2}|q)=}$ ${\displaystyle q^{(-\alpha /2-1/4)*n}*P_{n}^{(}\alpha )(cos\theta |q)}$ 

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${\displaystyle x=q^{x}}$ ,q→1,即得拉盖尔多项式

验证

3阶连续q拉盖尔多项式： ${\displaystyle \lim _{q\to 1}P_{3}^{(}a)=1/6\,{a}^{3}-x{a}^{2}+{a}^{2}+{\frac {11}{6}}\,a+2\,a{x}^{2}-5\,ax+1-6\,x-4/3\,{x}^{3}+6\,{x}^{2}}$ 

3阶广义拉盖尔多项式：

${\displaystyle L_{3}^{a}(2x)={\frac {1}{6}}(a+1)_{3}*_{1}F_{1}(-n,a+1;2x)}$  ${\displaystyle =1/6\,{a}^{3}-x{a}^{2}+{a}^{2}+{\frac {11}{6}}\,a+2\,a{x}^{2}-5\,ax+1-6\,x-4/3\,{x}^{3}+6\,{x}^{2}}$ 

两者显然相等。

图集

CONTINUOUS Q LAGUERRE ABS COMPLEX 3D MAPLE PLOT

CONTINUOUS Q LAGUERRE IM COMPLEX 3D MAPLE PLOT

CONTINUOUS Q LAGUERRE RE COMPLEX 3D MAPLE PLOT

CONTINUOUS Q LAGUERRE ABS DENSITY MAPLE PLOT

CONTINUOUS Q LAGUERRE IM DENSITY MAPLE PLOT

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 |contribution-url=缺少标题 (帮助), 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 

1. Roelof Koekoek, Peter Lesky, Rene Swarttouw,Hypergeometric Orthogonal Polynomials and Their q-Analogues, p514, Springer