Weber’s equation and hermite polynomials – HP 48gII User Manual
Page 537
Page 16-59
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.
L
2
(x) = 1-2x+ 0.5x
2
L
3
(x) = 1-3x+1.5x
2
-0.16666…x
3
.
Weber’s equation and Hermite polynomials
Weber’s equation is defined as d
2
y/dx
2
+(n+1/2-x
2
/4)y = 0, for n = 0, 1,
2, … A particular solution of this equation is given by the function , y(x) =
exp(-x
2
/4)H
*
(x/
√2), where the function H
*
(x) is the Hermite polynomial:
,..
2
,
1
),
(
)
1
(
)
(
*
,
1
*
2
2
0
=
−
=
=
−
n
e
dx
d
e
x
H
H
x
n
n
x
n
n
In the calculator, the function HERMITE, available through the menu
ARITHMETIC/POLYNOMIAL. Function HERMITE takes as argument an integer
number, n, and returns the Hermite polynomial of n-th degree. For example,
the first four Hermite polynomials are obtained by using: