HP 50G, User Manual

Page: 533 / 887

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
=
−
=
⎟⎟⎠
⎞
⎜⎜⎝
⎛