Laguerre’s equation, Laguerre’s equation ,16-56 – HP 50g Graphing Calculator User Manual
Page 533
Page 16-56
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
,
are solutions to Laguerre’s equation. Laguerre’s polynomials can also be
calculated with:
The term
is the m-th coefficient of the binomial expansion (x+y)
n
. It also represents the
number of combinations of n elements taken m at a time. This function is
available in the calculator as function COMB in the MTH/PROB menu (see also
Chapter 17).
You can define the following function to calculate Laguerre’s polynomials:
When done typing it in the equation writer press use function DEFINE to create
the function L(x,n) into variable
@@@L@@@ .
To generate the first four Laguerre polynomials use, L(x,0), L(x,1), L(x,2), L(x,3).
The results are:
L
0
(x) = .
L
1
(x) = 1-x.
,...
2
,
1
,
)
(
!
)
(
,
1
)
(
0
=
⋅
⋅
=
=
−
n
dx
e
x
d
n
e
x
L
x
L
n
x
n
n
x
n
.
!
)
1
(
)
(
0
m
n
m
m
n
x
m
n
m
x
L
⋅
⎟⎟⎠
⎞
⎜⎜⎝
⎛
⋅
−
=
∑
=
n
n
x
n
x
n
n
x
n
⋅
−
+
+
−
⋅
−
+
⋅
−
=
!
)
1
(
....
...
4
)
1
(
1
2
)
,
(
)!
(
!
!
m
n
C
m
n
m
n
m
n
=
−
=
⎟⎟⎠
⎞
⎜⎜⎝
⎛