Q拉盖尔多项式

Q-拉盖尔多项式满足下列正交关系

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

小q雅可比多项式→Q拉盖尔多项式.

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

Q梅西纳多项式→Q拉盖尔多项式;

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

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

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

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