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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
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
Figure 18.4.1:
Jacobi polynomials
${P}_{n}^{(1.5,-0.5)}\left(x\right)$
,
$n=1,2,3,4,5$
.
Symbols:
${P}_{n}^{(\alpha ,\beta )}\left(x\right)$
: Jacobi polynomial
,
$n$
: nonnegative integer
and
$x$
: real variable
Keywords:
Jacobi polynomials
Referenced by:
§18.2(vi)
Permalink:
http://dlmf.nist.gov/18.4.F1
Encodings:
pdf
,
png
Figure 18.4.2:
Jacobi polynomials
${P}_{n}^{(1.25,0.75)}\left(x\right)$
,
$n=7,8$
. This illustrates inequalities for extrema of a Jacobi polynomial; see (
18.14.16
). See also
Askey (
1990
)
.
Symbols:
${P}_{n}^{(\alpha ,\beta )}\left(x\right)$
: Jacobi polynomial
,
$n$
: nonnegative integer
and
$x$
: real variable
Keywords:
Jacobi polynomials
Permalink:
http://dlmf.nist.gov/18.4.F2
Encodings:
pdf
,
png
Figure 18.4.3:
Chebyshev polynomials
${T}_{n}\left(x\right)$
,
$n=1,2,3,4,5$
.
Symbols:
${T}_{n}\left(x\right)$
: Chebyshev polynomial of the first kind
,
$n$
: nonnegative integer
and
$x$
: real variable
Keywords:
Chebyshev polynomials
Referenced by:
§3.11(ii)
Permalink:
http://dlmf.nist.gov/18.4.F3
Encodings:
pdf
,
png
Figure 18.4.4:
Legendre polynomials
${P}_{n}\left(x\right)$
,
$n=1,2,3,4,5$
.
Symbols:
${P}_{n}\left(x\right)$
: Legendre polynomial
,
$n$
: nonnegative integer
and
$x$
: real variable
Keywords:
Legendre polynomials
Permalink:
http://dlmf.nist.gov/18.4.F4
Encodings:
pdf
,
png
Figure 18.4.5:
Laguerre polynomials
${L}_{n}\left(x\right)$
,
$n=1,2,3,4,5$
.
Symbols:
${L}_{n}^{(\alpha )}\left(x\right)$
: Laguerre (or generalized Laguerre) polynomial
,
$n$
: nonnegative integer
and
$x$
: real variable
Keywords:
Laguerre polynomials
Permalink:
http://dlmf.nist.gov/18.4.F5
Encodings:
pdf
,
png
Figure 18.4.6:
Laguerre polynomials
${L}_{3}^{(\alpha )}\left(x\right)$
,
$\alpha =0,1,2,3,4$
.
Symbols:
${L}_{n}^{(\alpha )}\left(x\right)$
: Laguerre (or generalized Laguerre) polynomial
and
$x$
: real variable
Keywords:
Laguerre polynomials
Permalink:
http://dlmf.nist.gov/18.4.F6
Encodings:
pdf
,
png
Figure 18.4.7:
Monic Hermite polynomials
${h}_{n}(x)={2}^{-n}{H}_{n}\left(x\right)$
,
$n=1,2,3,4,5$
.
Symbols:
${H}_{n}\left(x\right)$
: Hermite polynomial
,
$n$
: nonnegative integer
and
$x$
: real variable
Keywords:
Hermite polynomials
Referenced by:
§18.2(vi)
Permalink:
http://dlmf.nist.gov/18.4.F7
Encodings:
pdf
,
png
§18.4(ii)
Surfaces
Notes:
These surfaces were produced at NIST.
Permalink:
http://dlmf.nist.gov/18.4.ii
Visualization Help
Figure 18.4.8:
Laguerre polynomials
${L}_{3}^{(\alpha )}\left(x\right)$
,
$0\le \alpha \le 3$
,
$0\le x\le 10$
.
Symbols:
${L}_{n}^{(\alpha )}\left(x\right)$
: Laguerre (or generalized Laguerre) polynomial
and
$x$
: real variable
Permalink:
http://dlmf.nist.gov/18.4.F8
Encodings:
VRML
,
X3D
,
pdf
,
png
Visualization Help
Figure 18.4.9:
Laguerre polynomials
${L}_{4}^{(\alpha )}\left(x\right)$
,
$0\le \alpha \le 3$
,
$0\le x\le 10$
.
Symbols:
${L}_{n}^{(\alpha )}\left(x\right)$
: Laguerre (or generalized Laguerre) polynomial
and
$x$
: real variable
Permalink:
http://dlmf.nist.gov/18.4.F9
Encodings:
VRML
,
X3D
,
pdf
,
png