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Orthogonal Polynomials
Orthogonal polynomials areemployed in many areasofmathematics and numerical
analysis. They have been studied thoroughly andmany of their properties areknown.
What follows is avery small compendium of a largetopic.
The polynomials
(
n
is the degree of the polynomial) aresaid
to form an
orthogonal set
in the interval(
a
ϕ
n
(
x
),
n
=
0
,
1
,
2
,...
,
b
) with respect to the
weighting function
w
(
x
) if
b
w
(
x
)
ϕ
m
(
x
)
ϕ
n
(
x
)
dx
=
0,
m
=
n
(6.18)
a
The set is determined,exceptforaconstantfactor, by the choice of the weighting func-
tion and the limits of integration. That is, each set of orthogonal polynomials is asso-
ciatedwith certain
w
(
x
),
a
and
b
. The constantfactorisspecifiedbystandardization.
Some of the classicalorthogonal polynomials, namedafterwell-known mathemati-
cians, arelistedinTable 6.1. The last column in the table shows the standardization
used.
a
w
(
x
)
ϕ
n
(
x
)
2
dx
Name
Symbol
a
b
w
(
x
)
Legendre
p
n
(
x
)
−
1
1
1
2
/
(2
n
+
1)
x
2
)
−
1
/
2
Chebyshev
T
n
(
x
)
−
1
1
(1
−
π/
2 (
n
>
0)
e
−
x
Laguerre
L
n
(
x
)
0
∞
1
√
π
e
−
x
2
2
n
n
!
Hermite
H
n
(
x
)
−∞
∞
Table 6.1
Orthogonal polynomials obeyrecurrence relations of the form
a
n
ϕ
n
+
1
(
x
)
=
(
b
n
+
c
n
x
)
ϕ
n
(
x
)
−
d
n
ϕ
n
−
1
(
x
)
(6.19)
If the first two polynomials of the set areknown, the othermembers of the set can be
computed fromEq. (6.19). The coefficients in the recurrence formula,togetherwith
ϕ
0
(
x
) and
ϕ
1
(
x
), are giveninTable 6.2.
Name
ϕ
0
(
x
)
ϕ
1
(
x
)
a
n
b
n
c
n
d
n
+
+
Legendre
1
x
n
1
0
2
n
1
n
Chebyshev
1
x
1
0
2
1
Laguerre
1
1
−
x
n
+
1
2
n
+
1
−
1
n
Hermite
1
2
x
1
0
2
2
Table 6.2
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