Q拉盖尔多项式

q拉盖尔多项式是一个以基本超几何函数和Q阶乘幂定义的正交多项式

\displaystyle L_{n}^{(\alpha )}(x;q)={\frac {(q^{\alpha +1};q)_{n}}{(q;q)_{n}}}{}_{1}\phi _{1}(q^{-n};q^{\alpha +1};q,-q^{n+\alpha +1}x)

正交性

Q-拉盖尔多项式满足下列正交关系

\int _{a}^{b}\!L_{n}^{(\alpha )}(x;q)*L_{m}^{(\alpha )}(x;q)*(x^{\alpha })/(-x;q)_{\infty }\,dx={\frac {(q^{\alpha }+1;q)_{n}}{(q;q)_{n}*q^{n}}}

在校q雅可比多项式的定义中，令 $a=q^{\alpha }$ 以及 $x\to -b^{-1}q^{-1}$ ,并令 $b\to \infty$ ,即得q拉盖尔多项式。

令Q梅西纳多项式中 $b=q^{\alpha }$ ,以及 $q^{-x}=cq^{\alpha }x$ ,然后取 $c\to \infty$ 即得Q拉盖尔多项式。

$\lim _{c\to \infty }M_{n}(cq^{\alpha }x;q^{\alpha },c;q)={\frac {(q;q)_{n}}{(q^{\alpha +1};q)_{n}}}$

下列 : $\displaystyle L_{5}^{(3.7)}(x+iy;q)$ 图，以q 为可变参数。

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