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18
Orthogonal Polynomials
Classical Orthogonal Polynomials
18.3
Definitions
18.5
Explicit Representations
§18.4
Graphics
ⓘ
Permalink:
http://dlmf.nist.gov/18.4
See also:
Annotations for
Ch.18
Contents
§18.4(i)
Graphs
§18.4(ii)
Surfaces
§18.4(i)
Graphs
ⓘ
Notes:
These graphs were produced at NIST.
Permalink:
http://dlmf.nist.gov/18.4.i
See also:
Annotations for
§18.4
and
Ch.18
Figure 18.4.1:
Jacobi polynomials
P
n
(
1.5
,
−
0.5
)
(
x
)
,
n
=
1
,
2
,
3
,
4
,
5
.
Magnify
ⓘ
Symbols:
P
n
(
α
,
β
)
(
x
)
: Jacobi polynomial
,
n
: nonnegative integer
and
x
: real variable
Keywords:
Jacobi polynomials
,
graphs
Referenced by:
§18.2(vi)
Permalink:
http://dlmf.nist.gov/18.4.F1
Encodings:
pdf
,
png
See also:
Annotations for
§18.4(i)
,
§18.4
and
Ch.18
Figure 18.4.2:
Jacobi polynomials
P
n
(
1.25
,
0.75
)
(
x
)
,
n
=
7
,
8
. This illustrates inequalities for extrema of a Jacobi polynomial; see (
18.14.16
). See also
Askey (
1990
)
.
Magnify
ⓘ
Symbols:
P
n
(
α
,
β
)
(
x
)
: Jacobi polynomial
,
n
: nonnegative integer
and
x
: real variable
Keywords:
Jacobi polynomials
,
graphs
Permalink:
http://dlmf.nist.gov/18.4.F2
Encodings:
pdf
,
png
See also:
Annotations for
§18.4(i)
,
§18.4
and
Ch.18
Figure 18.4.3:
Chebyshev polynomials
T
n
(
x
)
,
n
=
1
,
2
,
3
,
4
,
5
.
Magnify
ⓘ
Symbols:
T
n
(
x
)
: Chebyshev polynomial of the first kind
,
n
: nonnegative integer
and
x
: real variable
Keywords:
Chebyshev polynomials
,
graphs
Referenced by:
§3.11(ii)
Permalink:
http://dlmf.nist.gov/18.4.F3
Encodings:
pdf
,
png
See also:
Annotations for
§18.4(i)
,
§18.4
and
Ch.18
Figure 18.4.4:
Legendre polynomials
P
n
(
x
)
,
n
=
1
,
2
,
3
,
4
,
5
.
Magnify
ⓘ
Symbols:
P
n
(
x
)
: Legendre polynomial
,
n
: nonnegative integer
and
x
: real variable
Keywords:
Legendre polynomials
,
graphs
Permalink:
http://dlmf.nist.gov/18.4.F4
Encodings:
pdf
,
png
See also:
Annotations for
§18.4(i)
,
§18.4
and
Ch.18
Figure 18.4.5:
Laguerre polynomials
L
n
(
x
)
,
n
=
1
,
2
,
3
,
4
,
5
.
Magnify
ⓘ
Symbols:
L
n
(
x
)
=
L
n
(
0
)
(
x
)
: Laguerre polynomial
,
n
: nonnegative integer
and
x
: real variable
Keywords:
Laguerre polynomials
,
graphics
Permalink:
http://dlmf.nist.gov/18.4.F5
Encodings:
pdf
,
png
See also:
Annotations for
§18.4(i)
,
§18.4
and
Ch.18
Figure 18.4.6:
Laguerre polynomials
L
3
(
α
)
(
x
)
,
α
=
0
,
1
,
2
,
3
,
4
.
Magnify
ⓘ
Symbols:
L
n
(
α
)
(
x
)
: Laguerre (or generalized Laguerre) polynomial
and
x
: real variable
Keywords:
Laguerre polynomials
,
graphics
Permalink:
http://dlmf.nist.gov/18.4.F6
Encodings:
pdf
,
png
See also:
Annotations for
§18.4(i)
,
§18.4
and
Ch.18
Figure 18.4.7:
Monic Hermite polynomials
h
n
(
x
)
=
2
−
n
H
n
(
x
)
,
n
=
1
,
2
,
3
,
4
,
5
.
Magnify
ⓘ
Symbols:
H
n
(
x
)
: Hermite polynomial
,
n
: nonnegative integer
and
x
: real variable
Keywords:
Hermite polynomials
,
graphs
,
monic
Referenced by:
§18.2(vi)
Permalink:
http://dlmf.nist.gov/18.4.F7
Encodings:
pdf
,
png
See also:
Annotations for
§18.4(i)
,
§18.4
and
Ch.18
§18.4(ii)
Surfaces
ⓘ
Notes:
These surfaces were produced at NIST.
Permalink:
http://dlmf.nist.gov/18.4.ii
See also:
Annotations for
§18.4
and
Ch.18
Figure 18.4.8:
Laguerre polynomials
L
3
(
α
)
(
x
)
,
0
≤
α
≤
3
,
0
≤
x
≤
10
.
Magnify
3D
Help
ⓘ
Symbols:
L
n
(
α
)
(
x
)
: Laguerre (or generalized Laguerre) polynomial
and
x
: real variable
Permalink:
http://dlmf.nist.gov/18.4.F8
Encodings:
Vizualization
,
pdf
,
png
See also:
Annotations for
§18.4(ii)
,
§18.4
and
Ch.18
Figure 18.4.9:
Laguerre polynomials
L
4
(
α
)
(
x
)
,
0
≤
α
≤
3
,
0
≤
x
≤
10
.
Magnify
3D
Help
ⓘ
Symbols:
L
n
(
α
)
(
x
)
: Laguerre (or generalized Laguerre) polynomial
and
x
: real variable
Permalink:
http://dlmf.nist.gov/18.4.F9
Encodings:
Vizualization
,
pdf
,
png
See also:
Annotations for
§18.4(ii)
,
§18.4
and
Ch.18