Laguerre’s equation – HP 48gII User Manual
Page 536
Page 16-58
U
n
(x) = sin(n
⋅arccos(x))/sin(arccos(x)).
You can access the function TCHEBYCHEFF through the command catalog
(
‚N).
The first four Chebyshev or Tchebycheff polynomials of the first and second
kind are obtained as follows:
0 TCHEBYCHEFF, result: 1,
i.e.,
T
0
(x) = 1.0.
-0 TCHEBYCHEFF, result: 1,
i.e.,
U
0
(x) = 1.0.
1 TCHEBYCHEFF, result: ‘X’,
i.e.,
T
1
(x) = x.
-1 TCHEBYCHEFF, result: 1,
i.e.,
U
1
(x) =1.0.
2 TCHEBYCHEFF, result: ‘2*X^2-1,
i.e.,
T
2
(x) =2x
2
-1.
-2 TCHEBYCHEFF, result: ‘2*X’,
i.e.,
U
2
(x) =2x.
3 TCHEBYCHEFF, result: ‘4*X^3-3*X’, i.e.,
T
3
(x) = 4x
3
-3x.
-3 TCHEBYCHEFF, result: ‘4*X^2-1’,
i.e.,
U
3
(x) = 4x
2
-1.
Laguerre’s equation
Laguerre’s equation is the second-order, linear ODE of the form x
⋅(d
2
y/dx
2
)
+(1
−x)⋅ (dy/dx) + n⋅y = 0. Laguerre polynomials, defined as
,...
2
,
1
,
)
(
!
)
(
,
1
)
(
0
=
⋅
⋅
=
=
−
n
dx
e
x
d
n
e
x
L
x
L
n
x
n
n
x
n
,
are solutions to Laguerre’s equation. Laguerre’s polynomials can also be
calculated with:
.
!
)
1
(
)
(
0
m
n
m
m
n
x
m
n
m
x
L
⋅
⋅
−
=
∑
=
The term
)
,
(
)!
(
!
!
m
n
C
m
n
m
n
m
n
=
−
=
n
n
x
n
x
n
n
x
n
⋅
−
+
+
−
⋅
−
+
⋅
−
=
!
)
1
(
....
...
4
)
1
(
1
2