大Q勒让德多项式

**大 q - 勒讓德多項式**是一個以基本超幾何函數定義的正交多項式：

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

## 正交性

大 q - 勒讓德多項式滿足下列正交關係

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

## 極限關係

**大 Q 勒讓德多項式→勒讓德多項式**

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

 
令大 q 勒讓德項式中的 $c=0$ , 並且 q→1 即得勒讓德多項式

**驗證**

將 c=0 代人 7 階 (n=7) 大 q 勒讓德多項式得：

$$
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}
$$

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

另 7 階勒讓德多項式：

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

顯然 qL2=P\_7 (2x-1) QED.

## 圖集

下列複數域三階大 q 勒讓德多項式: $\displaystyle P_{n}3(x+iy;-1.5;q)$ 的

一組三個虛數部、實數部與絕對值的複數三維動畫圖，以 q 為可變參數

一組三個虛數部、實數部與絕對值的複數密度動畫圖

## 參考文獻

* Andrews, George E.; Askey, Richard, Classical orthogonal polynomials, Brezinski, C.; Draux, A.; Magnus, Alphonse P.; Maroni, Pascal; Ronveaux, A. (編), Polynômes orthogonaux et applications. Proceedings of the Laguerre symposium held at Bar-le-Duc, October 15–18, 1984., Lecture Notes in Math. **1171**, Berlin, New York: Springer-Verlag: 36–62, 1985, ISBN 978-3-540-16059-5, MR 0838970, doi:10.1007/BFb0076530
* Gasper, George; Rahman, Mizan, Basic hypergeometric series, Encyclopedia of Mathematics and its Applications **96** 2nd, Cambridge University Press, 2004, ISBN 978-0-521-83357-8, MR 2128719, doi:10.2277/0521833574
* Koekoek, Roelof; Lesky, Peter A.; Swarttouw, René F., Hypergeometric orthogonal polynomials and their q-analogues, Springer Monographs in Mathematics, Berlin, New York: Springer-Verlag, 2010, ISBN 978-3-642-05013-8, MR 2656096, doi:10.1007/978-3-642-05014-5
* Koornwinder, Tom H.; Wong, Roderick S. C.; Koekoek, Roelof; Swarttouw, René F., http://dlmf.nist.gov/18, Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (編), NIST Handbook of Mathematical Functions, Cambridge University Press, 2010, ISBN 978-0521192255, MR2723248