- 1
- 2
- Dual Hahn polynomials
- Hahn polynomials
- Continuous dual Hahn polynomials
- Continuous Hahn polynomials
- Orthogonal polynomials
- Dual q-Hahn polynomials
- Discrete orthogonal polynomials
- Continuous dual q-Hahn polynomials
- List of dualities
- Askey scheme
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In mathematics, the dual Hahn polynomials are a family of orthogonal polynomials in the Askey scheme of hypergeometric orthogonal polynomials. They are defined on a non-uniform lattice
x
(
s
)
=
s
(
s
+
1
)
{\displaystyle x(s)=s(s+1)}
and are defined as
w
n
(
c
)
(
s
,
a
,
b
)
=
(
a
−
b
+
1
)
n
(
a
+
c
+
1
)
n
n
!
3
F
2
(
−
n
,
a
−
s
,
a
+
s
+
1
;
a
−
b
+
a
,
a
+
c
+
1
;
1
)
{\displaystyle w_{n}^{(c)}(s,a,b)={\frac {(a-b+1)_{n}(a+c+1)_{n}}{n!}}{}_{3}F_{2}(-n,a-s,a+s+1;a-b+a,a+c+1;1)}
for
n
=
0
,
1
,
.
.
.
,
N
−
1
{\displaystyle n=0,1,...,N-1}
and the parameters
a
,
b
,
c
{\displaystyle a,b,c}
are restricted to
−
1
2
<
a
<
b
,
|
c
|
<
1
+
a
,
b
=
a
+
N
{\displaystyle -{\frac {1}{2}}
.
Note that
(
u
)
k
{\displaystyle (u)_{k}}
is the rising factorial, otherwise known as the Pochhammer symbol, and
3
F
2
(
⋅
)
{\displaystyle {}_{3}F_{2}(\cdot )}
is the generalized hypergeometric functions
Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw (2010, 14) give a detailed list of their properties.
Orthogonality
The dual Hahn polynomials have the orthogonality condition
∑
s
=
a
b
−
1
w
n
(
c
)
(
s
,
a
,
b
)
w
m
(
c
)
(
s
,
a
,
b
)
ρ
(
s
)
[
Δ
x
(
s
−
1
2
)
]
=
δ
n
m
d
n
2
{\displaystyle \sum _{s=a}^{b-1}w_{n}^{(c)}(s,a,b)w_{m}^{(c)}(s,a,b)\rho (s)[\Delta x(s-{\frac {1}{2}})]=\delta _{nm}d_{n}^{2}}
for
n
,
m
=
0
,
1
,
.
.
.
,
N
−
1
{\displaystyle n,m=0,1,...,N-1}
. Where
Δ
x
(
s
)
=
x
(
s
+
1
)
−
x
(
s
)
{\displaystyle \Delta x(s)=x(s+1)-x(s)}
,
ρ
(
s
)
=
Γ
(
a
+
s
+
1
)
Γ
(
c
+
s
+
1
)
Γ
(
s
−
a
+
1
)
Γ
(
b
−
s
)
Γ
(
b
+
s
+
1
)
Γ
(
s
−
c
+
1
)
{\displaystyle \rho (s)={\frac {\Gamma (a+s+1)\Gamma (c+s+1)}{\Gamma (s-a+1)\Gamma (b-s)\Gamma (b+s+1)\Gamma (s-c+1)}}}
and
d
n
2
=
Γ
(
a
+
c
+
n
+
a
)
n
!
(
b
−
a
−
n
−
1
)
!
Γ
(
b
−
c
−
n
)
.
{\displaystyle d_{n}^{2}={\frac {\Gamma (a+c+n+a)}{n!(b-a-n-1)!\Gamma (b-c-n)}}.}
Numerical instability
As the value of
n
{\displaystyle n}
increases, the values that the discrete polynomials obtain also increases. As a result, to obtain numerical stability in calculating the polynomials you would use the renormalized dual Hahn polynomial as defined as
w
^
n
(
c
)
(
s
,
a
,
b
)
=
w
n
(
c
)
(
s
,
a
,
b
)
ρ
(
s
)
d
n
2
[
Δ
x
(
s
−
1
2
)
]
{\displaystyle {\hat {w}}_{n}^{(c)}(s,a,b)=w_{n}^{(c)}(s,a,b){\sqrt {{\frac {\rho (s)}{d_{n}^{2}}}[\Delta x(s-{\frac {1}{2}})]}}}
for
n
=
0
,
1
,
.
.
.
,
N
−
1
{\displaystyle n=0,1,...,N-1}
.
Then the orthogonality condition becomes
∑
s
=
a
b
−
1
w
^
n
(
c
)
(
s
,
a
,
b
)
w
^
m
(
c
)
(
s
,
a
,
b
)
=
δ
m
,
n
{\displaystyle \sum _{s=a}^{b-1}{\hat {w}}_{n}^{(c)}(s,a,b){\hat {w}}_{m}^{(c)}(s,a,b)=\delta _{m,n}}
for
n
,
m
=
0
,
1
,
.
.
.
,
N
−
1
{\displaystyle n,m=0,1,...,N-1}
Relation to other polynomials
The Hahn polynomials,
h
n
(
x
,
N
;
α
,
β
)
{\displaystyle h_{n}(x,N;\alpha ,\beta )}
, is defined on the uniform lattice
x
(
s
)
=
s
{\displaystyle x(s)=s}
, and the parameters
a
,
b
,
c
{\displaystyle a,b,c}
are defined as
a
=
(
α
+
β
)
/
2
,
b
=
a
+
N
,
c
=
(
β
−
α
)
/
2
{\displaystyle a=(\alpha +\beta )/2,b=a+N,c=(\beta -\alpha )/2}
. Then setting
α
=
β
=
0
{\displaystyle \alpha =\beta =0}
the Hahn polynomials become the Chebyshev polynomials. Note that the dual Hahn polynomials have a q-analog with an extra parameter q known as the dual q-Hahn polynomials.
Racah polynomials are a generalization of dual Hahn polynomials.
References
Zhu, Hongqing (2007), "Image analysis by discrete orthogonal dual Hahn moments" (PDF), Pattern Recognition Letters, 28 (13): 1688–1704, doi:10.1016/j.patrec.2007.04.013
Hahn, Wolfgang (1949), "Über Orthogonalpolynome, die q-Differenzengleichungen genügen", Mathematische Nachrichten, 2 (1–2): 4–34, doi:10.1002/mana.19490020103, ISSN 0025-584X, MR 0030647
Koekoek, Roelof; Lesky, Peter A.; Swarttouw, René F. (2010), Hypergeometric orthogonal polynomials and their q-analogues, Springer Monographs in Mathematics, Berlin, New York: Springer-Verlag, doi:10.1007/978-3-642-05014-5, ISBN 978-3-642-05013-8, MR 2656096
Koornwinder, Tom H.; Wong, Roderick S. C.; Koekoek, Roelof; Swarttouw, René F. (2010), "Hahn Class: Definitions", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248.
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