連續q拉蓋爾多項式

3rd order Continuous q Laguerre polynomials

連續q拉蓋爾多項式(Continuous q-Laguerre polynomials)是一個以基本超幾何函數定義的正交多項式。

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

極限關係

Q梅西納-帕拉澤克多項式→連續q拉蓋爾多項式

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

阿拉-薩拉姆-遲哈剌多項式→連續q拉蓋爾多項式

令連續q拉蓋爾多項式中$x=q^{x}$,q→1,即得拉蓋爾多項式

驗證

3階連續q拉蓋爾多項式： $\lim _{q\to 1}P_{3}^{(}a)=1/6\,{a}^{3}-x{a}^{2}+{a}^{2}+{\frac {11}{6}}\,a+2\,a{x}^{2}-5\,ax+1-6\,x-4/3\,{x}^{3}+6\,{x}^{2}$

3階廣義拉蓋爾多項式：

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

兩者顯然相等。

圖集

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

| |}

參考文獻

• 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. (編), NIST Handbook of Mathematical Functions, Cambridge University Press, 2010, ISBN 978-0521192255, MR2723248