大Q勒让德多项式

大q-勒让德多项式是一个以基本超几何函数定义的正交多项式^[1]：

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${\displaystyle \displaystyle P_{n}(x;c;q)={}_{3}\phi _{2}(q^{-n},q^{n+1},x;q,cq;q,q)}$

正交性

大q-勒让德多项式满足下列正交关系

${\displaystyle \int _{cq}^{q}P_{m}(x;c;q)P_{n}(x;c;q)d_{q}x=q(1-c){\frac {1-q}{1-q^{2n+1}}}{\frac {(c^{-1}q;q)_{n}}{(cq;q)_{n}}}(-cq^{2})^{n}q^{n \choose 2}\delta _{mn}}$

极限关系

大Q勒让德多项式→勒让德多项式

${\displaystyle \displaystyle \lim _{q\to 1}P_{n}(x;0;q)=P_{n}(2x-1)}$

令大q勒让德项式中的${\displaystyle c=0}$,并且q→1 即得勒让德多项式

验证

将c=0代人7阶(n=7)大q勒让德多项式得：

${\displaystyle qL=P_{n}(x;0;q)=1+{\frac {q}{\left(1-q\right)^{2}}}-{\frac {qx}{\left(1-q\right)^{2}}}-{\frac {{q}^{9}}{\left(1-q\right)^{2}}}+{\frac {x{q}^{9}}{\left(1-q\right)^{2}}}-{\frac {1}{{q}^{6}\left(1-q\right)^{2}}}+{\frac {x}{{q}^{6}\left(1-q\right)^{2}}}+{\frac {{q}^{2}}{\left(1-q\right)^{2}}}-{\frac {x{q}^{2}}{\left(1-q\right)^{2}}}+\left(1-{q}^{-7}\right)\left(1-{q}^{-6}\right)\left(1-{q}^{8}\right)\left(1-{q}^{9}\right)\left(1-x\right)\left(1-qx\right){q}^{2}\left(1-q\right)^{-2}\left(1-{q}^{2}\right)^{-2}+\left(1-{q}^{-7}\right)\left(1-{q}^{-6}\right)\left(1-{q}^{-5}\right)\left(1-{q}^{8}\right)\left(1-{q}^{9}\right)\left(1-{q}^{10}\right)\left(1-x\right)\left(1-qx\right)\left(1-x{q}^{2}\right){q}^{3}\left(1-q\right)^{-2}\left(1-{q}^{2}\right)^{-2}\left(1-{q}^{3}\right)^{-2}+\left(1-{q}^{-7}\right)\left(1-{q}^{-6}\right)\left(1-{q}^{-5}\right)\left(1-{q}^{-4}\right)\left(1-{q}^{8}\right)\left(1-{q}^{9}\right)\left(1-{q}^{10}\right)\left(1-{q}^{11}\right)\left(1-x\right)\left(1-qx\right)\left(1-x{q}^{2}\right)\left(1-x{q}^{3}\right){q}^{4}\left(1-q\right)^{-2}\left(1-{q}^{2}\right)^{-2}\left(1-{q}^{3}\right)^{-2}\left(1-{q}^{4}\right)^{-2}+\left(1-{q}^{-7}\right)\left(1-{q}^{-6}\right)\left(1-{q}^{-5}\right)\left(1-{q}^{-4}\right)\left(1-{q}^{-3}\right)\left(1-{q}^{8}\right)\left(1-{q}^{9}\right)\left(1-{q}^{10}\right)\left(1-{q}^{11}\right)\left(1-{q}^{12}\right)\left(1-x\right)\left(1-qx\right)\left(1-x{q}^{2}\right)\left(1-x{q}^{3}\right)\left(1-x{q}^{4}\right){q}^{5}\left(1-q\right)^{-2}\left(1-{q}^{2}\right)^{-2}\left(1-{q}^{3}\right)^{-2}\left(1-{q}^{4}\right)^{-2}\left(1-{q}^{5}\right)^{-2}+\left(1-{q}^{-7}\right)\left(1-{q}^{-6}\right)\left(1-{q}^{-5}\right)\left(1-{q}^{-4}\right)\left(1-{q}^{-3}\right)\left(1-{q}^{-2}\right)\left(1-{q}^{8}\right)\left(1-{q}^{9}\right)\left(1-{q}^{10}\right)\left(1-{q}^{11}\right)\left(1-{q}^{12}\right)\left(1-{q}^{13}\right)\left(1-x\right)\left(1-qx\right)\left(1-x{q}^{2}\right)\left(1-x{q}^{3}\right)\left(1-x{q}^{4}\right)\left(1-x{q}^{5}\right){q}^{6}\left(1-q\right)^{-2}\left(1-{q}^{2}\right)^{-2}\left(1-{q}^{3}\right)^{-2}\left(1-{q}^{4}\right)^{-2}\left(1-{q}^{5}\right)^{-2}\left(1-{q}^{6}\right)^{-2}+\left(1-{q}^{-7}\right)\left(1-{q}^{-6}\right)\left(1-{q}^{-5}\right)\left(1-{q}^{-4}\right)\left(1-{q}^{-3}\right)\left(1-{q}^{-2}\right)\left(1-{q}^{-1}\right)\left(1-{q}^{8}\right)\left(1-{q}^{9}\right)\left(1-{q}^{10}\right)\left(1-{q}^{11}\right)\left(1-{q}^{12}\right)\left(1-{q}^{13}\right)\left(1-{q}^{14}\right)\left(1-x\right)\left(1-qx\right)\left(1-x{q}^{2}\right)\left(1-x{q}^{3}\right)\left(1-x{q}^{4}\right)\left(1-x{q}^{5}\right)\left(1-x{q}^{6}\right){q}^{7}\left(1-q\right)^{-2}\left(1-{q}^{2}\right)^{-2}\left(1-{q}^{3}\right)^{-2}\left(1-{q}^{4}\right)^{-2}\left(1-{q}^{5}\right)^{-2}\left(1-{q}^{6}\right)^{-2}\left(1-{q}^{7}\right)^{-2}}$

• ${\displaystyle qL2=\lim _{q\to 1}qL=-1+56\,x+3432\,{x}^{7}-12012\,{x}^{6}+16632\,{x}^{5}-11550\,{x}^{4}+4200\,{x}^{3}-756\,{x}^{2}}$

另7阶勒让德多项式：

• ${\displaystyle P_{7}(2x-1)=-1+56\,x+3432\,{x}^{7}-12012\,{x}^{6}+16632\,{x}^{5}-11550\,{x}^{4}+4200\,{x}^{3}-756\,{x}^{2}}$

显然qL2=P_7(2x-1) QED.

图集

下列复数域三阶大q勒让德多项式:${\displaystyle \displaystyle P_{n}3(x+iy;-1.5;q)}$的

一组三个虚数部、实数部与绝对值的复数三维动画图，以q为可变参数

一组三个虚数部、实数部与绝对值的复数密度动画图

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