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
4th order Little q-Laguerre polynomials
p
n
(
x
;
a
|
q
)
=
2
ϕ
1
(
q
−
n
,
0
;
a
q
;
q
,
q
x
)
=
1
(
a
−
1
q
−
n
;
q
)
n
2
ϕ
0
(
q
−
n
,
x
−
1
;
;
q
,
x
/
a
)
{\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)}
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
x
→
b
q
x
{\displaystyle x\to bqx}
b
→
∞
{\displaystyle b\to \infty }
lim
b
→
∞
P
n
(
b
q
x
;
a
,
b
;
q
)
=
p
n
(
x
;
a
|
q
)
{\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：
lim
a
→
1
=
K
n
a
f
f
(
q
x
−
N
;
p
,
N
|
q
)
=
p
n
(
q
x
;
p
,
q
)
{\displaystyle \lim _{a\to 1}=K_{n}^{aff}(q^{x-N};p,N|q)=p_{n}(q^{x};p,q)}
a
=
q
a
{\displaystyle a=q^{a}}
x
=
(
1
−
q
)
∗
x
{\displaystyle x=(1-q)*x}
拉蓋爾多項式
lim
q
→
1
P
a
(
1
−
q
)
x
;
q
a
|
q
)
=
L
n
(
a
)
(
x
)
L
n
(
a
)
(
0
)
{\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，
P
a
(
1
−
q
)
x
;
q
a
|
q
)
=
1
+
q
x
1
−
q
α
q
−
x
q
8
(
1
−
q
α
q
)
{\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)}}}
+
(
1
−
q
−
9
)
(
1
−
q
−
8
)
q
2
(
1
−
q
)
x
2
(
1
−
q
2
)
−
1
(
1
−
q
α
q
)
−
1
(
1
−
q
α
q
2
)
−
1
{\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}}
+
(
1
−
q
−
9
)
(
1
−
q
−
8
)
(
1
−
q
−
7
)
q
3
(
1
−
q
)
2
x
3
(
1
−
q
2
)
−
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}}
(
1
−
q
3
)
−
1
(
1
−
q
α
q
)
−
1
(
1
−
q
α
q
2
)
−
1
(
1
−
q
α
q
3
)
−
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 }
令a=3,得
(
1
−
9
4
x
+
9
5
x
2
−
7
10
x
3
+
3
20
x
4
−
3
160
x
5
+
1
720
x
6
−
1
16800
x
7
+
1
739200
x
8
−
1
79833600
x
9
)
{\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
L
n
(
3
)
(
x
)
L
n
(
3
)
(
0
)
{\displaystyle {\frac {L_{n}^{(3)}(x)}{L_{n}^{(3)}(0)}}}
(
1
−
9
4
x
+
9
5
x
2
−
7
10
x
3
+
3
20
x
4
−
3
160
x
5
+
1
720
x
6
−
1
16800
x
7
+
1
739200
x
8
−
1
79833600
x
9
)
{\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
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
Chihara, Theodore Seio, An introduction to orthogonal polynomials , Mathematics and its Applications 13 , New York: Gordon and Breach Science Publishers, 1978 [2015-02-06 ] , ISBN 978-0-677-04150-6MR 0481884 ISBN 978-0-486-47929-3 , （原始內容存檔 於2018-08-10） Gasper, George; Rahman, Mizan, Basic hypergeometric series, Encyclopedia of Mathematics and its Applications 96 2nd, Cambridge University Press , 2004, ISBN 978-0-521-83357-8MR 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 978-3-642-05013-8MR 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 978-0521192255MR 2723248 Van Assche, Walter; Koornwinder, Tom H., Asymptotic behaviour for Wall polynomials and the addition formula for little q-Legendre polynomials, SIAM Journal on Mathematical Analysis, 1991, 22 (1): 302–311, ISSN 0036-1410 MR 1080161 doi:10.1137/0522019 Wall, H. S., A continued fraction related to some partition formulas of Euler, The American Mathematical Monthly , 1941, 48 : 102–108, ISSN 0002-9890 JSTOR 2303599 MR 0003641 
