连续q-哈恩多项式 以基本超几何函数 定义:[1]
p
n
(
x
;
a
,
b
,
c
|
q
)
=
a
−
n
e
−
i
n
u
(
a
b
e
2
i
u
,
a
c
,
a
d
;
q
)
n
∗
4
Φ
3
(
q
−
n
,
a
b
c
d
q
n
−
1
,
a
e
i
t
+
2
u
,
a
e
−
i
t
;
a
b
e
2
i
u
,
a
c
,
a
d
;
q
;
q
)
{\displaystyle p_{n}(x;a,b,c|q)=a^{-n}e^{-inu}(abe^{2iu},ac,ad;q)_{n}*_{4}\Phi _{3}(q^{-n},abcdq^{n-1},ae^{i{t+2u}},ae^{-it};abe^{2iu},ac,ad;q;q)}
并且
x
=
c
o
s
(
t
+
u
)
{\displaystyle x=cos(t+u)}
极限关系
连续q哈恩多项式 →Q梅西纳-帕拉泽克多项式
p
n
(
c
o
s
(
θ
+
ϕ
)
;
a
,
0
,
0
,
a
;
q
)
(
q
;
q
)
n
=
P
n
(
c
o
s
(
θ
+
ϕ
)
;
a
|
q
)
{\displaystyle {\frac {p_{n}(cos(\theta +\phi );a,0,0,a;q)}{(q;q)_{n}}}=P_{n}(cos(\theta +\phi );a|q)}
阿斯基-威尔逊多项式 →连续q哈恩多项式
在阿斯基-威尔逊多项式中作代换
θ
→
θ
+
ϕ
{\displaystyle \theta \to \theta +\phi }
,
a
→
a
e
i
θ
{\displaystyle a\to ae^{i\theta }}
,
b
→
b
e
i
θ
{\displaystyle b\to be^{i\theta }}
,
c
→
c
e
−
i
θ
{\displaystyle c\to ce^{-i\theta }}
,
d
→
d
e
−
i
θ
{\displaystyle d\to de^{-i\theta }}
即得连续q哈恩多项式:
p
n
(
c
o
s
(
θ
+
ϕ
)
;
a
e
i
θ
,
b
e
i
θ
,
c
e
−
i
θ
,
d
e
−
i
θ
|
q
)
=
p
n
(
c
o
s
(
θ
+
ϕ
)
,
a
,
b
,
c
,
d
;
q
)
{\displaystyle p_{n}(cos(\theta +\phi );ae^{i\theta },be^{i\theta },ce^{-i\theta },de^{-i\theta }|q)=p_{n}(cos(\theta +\phi ),a,b,c,d;q)}
图集
CONTINUOUS q hahn ABS COMPLEX3D Maple PLOT
CONTINUOUS q hahn IIM COMPLEX3D Maple PLOT
CONTINUOUS q hahn RE COMPLEX3D Maple PLOT
CONTINUOUS q hahn ABS densitu Maple PLOT
CONTINUOUS q hahn im densitu Maple PLOT
CONTINUOUS q hahn RE densitu Maple PLOT
参考文献
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., Chapter 18: Orthogonal Polynomials , Olver, Frank W. J. ; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (编), NIST Handbook of Mathematical Functions , Cambridge University Press, 2010, ISBN 978-0521192255 , MR 2723248