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。

Q-LAGUERRE POLYNOMIALS ABS COMPLEX 3D MAPLE PLOT

Q-LAGUERRE POLYNOMIALS IM COMPLEX 3D MAPLE PLOT

Q-LAGUERRE POLYNOMIALS RE COMPLEX 3D MAPLE PLOT

Q-LAGUERRE POLYNOMIALS ABS DENSITY MAPLE PLOT

Q-LAGUERRE POLYNOMIALS RE DENSITY MAPLE PLOT

Q-LAGUERRE POLYNOMIALS IM DENSITY MAPLE PLOT

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