連續q拉蓋爾多項式

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^[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)}$

極限關係

Q梅西納-帕拉澤克多項式→連續q拉蓋爾多項式

${\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-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${\displaystyle x=q^{x}}$,q→1,即得拉蓋爾多項式

驗證

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

3階廣義拉蓋爾多項式：

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

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

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