小q拉蓋爾多項式

小q拉蓋爾多項式是一個以基本超幾何函數定義的正交多項式

${\displaystyle \displaystyle p_{n}(x;a|q)={}_{2}\phi _{1}(q^{-n},0;aq;q,qx)={\frac {1}{(a^{-1}q^{-n};q)_{n}}}{}_{2}\phi _{0}(q^{-n},x^{-1};;q,x/a)}$

4th order Little q-Laguerre polynomials

極限關係

大q拉蓋爾多項式→小q拉蓋爾多項式

在大q拉蓋爾多項式中，令${\displaystyle x\to bqx}$,並令${\displaystyle b\to \infty }$即得小q拉蓋爾多項式

${\displaystyle \lim _{b\to \infty }P_{n}(bqx;a,b;q)=p_{n}(x;a|q)}$

仿射Q克拉夫楚克多項式→ 小q拉蓋爾多項式：

${\displaystyle \lim _{a\to 1}=K_{n}^{aff}(q^{x-N};p,N|q)=p_{n}(q^{x};p,q)}$ 令小q拉蓋爾多項式 ${\displaystyle a=q^{a}}$ ${\displaystyle x=(1-q)*x}$,然後令q→1 即得拉蓋爾多項式

${\displaystyle \lim _{q\to 1}P_{a}(1-q)x;q^{a}|q)={\frac {L_{n}^{(a)}(x)}{L_{n}^{(a)}(0)}}}$

驗證 9階小q拉蓋爾多項式→拉蓋爾多項式

作上述代換，

${\displaystyle P_{a}(1-q)x;q^{a}|q)=1+{\frac {qx}{1-{q}^{\alpha }q}}-{\frac {x}{{q}^{8}\left(1-{q}^{\alpha }q\right)}}}$ ${\displaystyle +\left(1-{q}^{-9}\right)\left(1-{q}^{-8}\right){q}^{2}\left(1-q\right){x}^{2}\left(1-{q}^{2}\right)^{-1}\left(1-{q}^{\alpha }q\right)^{-1}\left(1-{q}^{\alpha }{q}^{2}\right)^{-1}}$ ${\displaystyle +\left(1-{q}^{-9}\right)\left(1-{q}^{-8}\right)\left(1-{q}^{-7}\right){q}^{3}\left(1-q\right)^{2}{x}^{3}\left(1-{q}^{2}\right)^{-1}}$ ${\displaystyle \left(1-{q}^{3}\right)^{-1}\left(1-{q}^{\alpha }q\right)^{-1}\left(1-{q}^{\alpha }{q}^{2}\right)^{-1}\left(1-{q}^{\alpha }{q}^{3}\right)^{-1}+\cdots }$

求q→1極限得

令a=3,得 ${\displaystyle (1-{\frac {9}{4}}x+{\frac {9}{5}}{x}^{2}-{\frac {7}{10}}{x}^{3}+{\frac {3}{20}}{x}^{4}-{\frac {3}{160}}{x}^{5}+{\frac {1}{720}}{x}^{6}-{\frac {1}{16800}}{x}^{7}+{\frac {1}{739200}}{x}^{8}-{\frac {1}{79833600}}{x}^{9})}$

另一方面

${\displaystyle {\frac {L_{n}^{(3)}(x)}{L_{n}^{(3)}(0)}}}$ ${\displaystyle =(1-{\frac {9}{4}}x+{\frac {9}{5}}{x}^{2}-{\frac {7}{10}}{x}^{3}+{\frac {3}{20}}{x}^{4}-{\frac {3}{160}}{x}^{5}+{\frac {1}{720}}{x}^{6}-{\frac {1}{16800}}{x}^{7}+{\frac {1}{739200}}{x}^{8}-{\frac {1}{79833600}}{x}^{9})}$

二者顯然相等 QED

圖集

LITTLE Q-LAGUERRE POLYNOMIALS ABS COMPLEX 3D MAPLE PLOT

LITTLE Q-LAGUERRE POLYNOMIALS IM COMPLEX 3D MAPLE PLOT

LITTLE Q-LAGUERRE POLYNOMIALS RE COMPLEX 3D MAPLE PLOT

LITTLE Q-LAGUERRE POLYNOMIALS ABS DENSITY MAPLE PLOT

LITTLE Q-LAGUERRE POLYNOMIALS IM DENSITY MAPLE PLOT

LITTLE Q-LAGUERRE POLYNOMIALS RE DENSITY MAPLE PLOT

參考文獻

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