User's Manual

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
exd
n
e
xLxL
n
xnnx
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
xL ⋅








⋅
−
=
∑
=
The term
),(
)!(!
!
mnC
mnm
n
m
n
=
−
=








n
n
x
n
x
nn
xn ⋅
−
++−⋅
−
+⋅−=
!
)1(
.......
4
)1(
1
2