# Laguerre's Equation - HP 49g+ User Manual

Graphing calculator.

U
(x) = sin(n⋅arccos(x))/sin(arccos(x)).
n
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,
-0 TCHEBYCHEFF, result: 1,
1 TCHEBYCHEFF, result: 'X',
-1 TCHEBYCHEFF, result: 1,
2 TCHEBYCHEFF, result: '2*X^2-1,
-2 TCHEBYCHEFF, result: '2*X',
3 TCHEBYCHEFF, result: '4*X^3-3*X', i.e.,
-3 TCHEBYCHEFF, result: '4*X^2-1',

## Laguerre's equation

Laguerre's equation is the second-order, linear ODE of the form x⋅(d
+(1−x)⋅ (dy/dx) + n⋅y = 0. Laguerre polynomials, defined as
(
)
, 1
L
x
0
are solutions to Laguerre's equation. Laguerre's polynomials can also be
calculated with:
1
n
The term
i.e.,
i.e.,
i.e.,
i.e.,
i.e.,
i.e.,
i.e.,
x
n
n
(
e
d
x
(
)
L
x
n
!
n
dx
n
m
n
(
) 1
L
(
x
)
n
m
!
m
m
=
0
n
(
n
) 1
2
x
x
...
4
n
n
!
C
m
m
( !
n
m
)!
T
(x) = 1.0.
0
U
(x) = 1.0.
0
T
(x) = x.
1
U
(x) =1.0.
1
2
T
(x) =2x
-1.
2
U
(x) =2x.
2
3
T
(x) = 4x
-3x.
3
2
U
(x) = 4x
-1.
3
2
y/dx
x
)
e
,
1
2 ,
,...
n
,
n
m
x
.
n
(
) 1
n
....
x
n
!
(
n
,
m
)
Page 16-58
2
)