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NIST
18
Orthogonal Polynomials
Classical Orthogonal Polynomials
18.7
Interrelations and Limit Relations
18.9
Recurrence Relations and Derivatives
§18.8
Differential Equations
ⓘ
Notes:
For Table
18.8.1
, Rows 2, 3, 5, 9–10, 11–12, see
Szegő (
1975
, (4.2.4), (4.24.2), (4.7.5), (5.1.2), (5.5.2))
, respectively; Row 4 is the special case
α
=
β
of Row 2; Rows 6, 7, 8 are the special cases
α
=
β
=
-
1
2
,
α
=
β
=
1
2
,
α
=
β
=
0
, respectively, of Row 2.
Permalink:
http://dlmf.nist.gov/18.8
See also:
Annotations for
Ch.18
See Table
18.8.1
and also Table 22.6 of
Abramowitz and Stegun (
1964
)
.
Table 18.8.1:
Classical OP’s: differential equations
A
(
x
)
f
′′
(
x
)
+
B
(
x
)
f
′
(
x
)
+
C
(
x
)
f
(
x
)
+
λ
n
f
(
x
)
=
0
.
f
(
x
)
A
(
x
)
B
(
x
)
C
(
x
)
λ
n
P
n
(
α
,
β
)
(
x
)
1
-
x
2
β
-
α
-
(
α
+
β
+
2
)
x
0
n
(
n
+
α
+
β
+
1
)
(
sin
1
2
x
)
α
+
1
2
(
cos
1
2
x
)
β
+
1
2
×
P
n
(
α
,
β
)
(
cos
x
)
1
0
1
4
-
α
2
4
sin
2
1
2
x
+
1
4
-
β
2
4
cos
2
1
2
x
(
n
+
1
2
(
α
+
β
+
1
)
)
2
(
sin
x
)
α
+
1
2
P
n
(
α
,
α
)
(
cos
x
)
1
0
(
1
4
-
α
2
)
/
sin
2
x
(
n
+
α
+
1
2
)
2
C
n
(
λ
)
(
x
)
1
-
x
2
-
(
2
λ
+
1
)
x
0
n
(
n
+
2
λ
)
T
n
(
x
)
1
-
x
2
-
x
0
n
2
U
n
(
x
)
1
-
x
2
-
3
x
0
n
(
n
+
2
)
P
n
(
x
)
1
-
x
2
-
2
x
0
n
(
n
+
1
)
L
n
(
α
)
(
x
)
x
α
+
1
-
x
0
n
e
-
1
2
x
2
x
α
+
1
2
L
n
(
α
)
(
x
2
)
1
0
-
x
2
+
(
1
4
-
α
2
)
x
-
2
4
n
+
2
α
+
2
H
n
(
x
)
1
-
2
x
0
2
n
e
-
1
2
x
2
H
n
(
x
)
1
0
-
x
2
2
n
+
1
He
n
(
x
)
1
-
x
0
n
ⓘ
Symbols:
T
n
(
x
)
: Chebyshev polynomial of the first kind
,
U
n
(
x
)
: Chebyshev polynomial of the second kind
,
H
n
(
x
)
: Hermite polynomial
,
He
n
(
x
)
: Hermite polynomial
,
P
n
(
α
,
β
)
(
x
)
: Jacobi polynomial
,
L
n
(
α
)
(
x
)
: Laguerre (or generalized Laguerre) polynomial
,
P
n
(
x
)
: Legendre polynomial
,
cos
z
: cosine function
,
e
: base of natural logarithm
,
sin
z
: sine function
,
C
n
(
λ
)
(
x
)
: ultraspherical (or Gegenbauer) polynomial
,
n
: nonnegative integer
and
x
: real variable
Keywords:
Chebyshev polynomials
,
Hermite polynomials
,
Jacobi polynomials
,
Laguerre polynomials
,
Legendre polynomials
,
classical orthogonal polynomials
,
differential equation
,
differential equations
,
ultraspherical polynomials
A&S Ref:
22.6.1, 22.6.4, 22.6.5, 22.6.8, 22.6.15, 22.6.18, 22.6.19, 22.6.20, 22.6.21
Referenced by:
§18.8
,
§18.8
Permalink:
http://dlmf.nist.gov/18.8.T1
See also:
Annotations for
§18.8
and
Ch.18