Laguerre's Equation - HP 50g User Manual

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
L
0
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)
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:
( ) , 1 (
x L x
n
L ( x )
n
n ( n
1 n x
4
n
⎛ ⎞
⎜ ⎜ ⎟ ⎟
m m
⎝ ⎠
L (x) = .
0
L (x) = 1-x.
1
x n n
(
e d x
)
!
n dx
n
m
⎛
n ( ) 1
∑
⎜ ⎜
m
m !
⎝
m 0
) 1
2
x ...
n !
C
( ! n m )!
x
)
e
, 1 2 , ,...
n
n
⎞
m
⎟ ⎟
x .
⎠
n
( ) 1
n
.... x
n !
( n , m )
n
. It also represents the
2 2
y/dx )
,
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