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
P
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{\displaystyle P_{n}^{(\alpha )}(x|q)={\frac {(q^{\alpha }+1;q)_{n}}{(q;q)_{n}}}}
3
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{\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)}
Q梅西纳-帕拉泽克多项式 →连续q拉盖尔多项式
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{\displaystyle P_{n}(cos(\theta +\phi );q^{\alpha /2+1/2}|q)=}
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{\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
x
=
q
x
{\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:
lim
q
→
1
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{\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:
L
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{\displaystyle L_{3}^{a}(2x)={\frac {1}{6}}(a+1)_{3}*_{1}F_{1}(-n,a+1;2x)}
=
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{\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。
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 , MR 2723248
Roelof Koekoek, Peter Lesky, Rene Swarttouw,Hypergeometric Orthogonal Polynomials and Their q-Analogues, p514, Springer