HP 48gII - Manual (Page 191)

Page 5-20
The HERMITE function
The function HERMITE [HERMI] uses as argument an integer number, k, and
returns the Hermite polynomial of k-th degree. A Hermite polynomial, He
k
(x)
is defined as
,... 2 , 1 ), ( ) 1 ( ) ( , 1 2 / 2 /
0
2 2 = − = = −
n e
dx
d
e x He He
x
n
n
x n
n
An alternate definition of the Hermite polynomials is
,... 2 , 1 ), ( ) 1 ( ) ( * , 1 * 2 2
0
= − = = −
n e
dx
d
e x H H
x
n
n
x n
n
where d n /dx n = n-th derivative with respect to x. This is the definition used in
the calculator.
Examples: The Hermite polynomials of orders 3 and 5 are given by:
HERMITE(3) = ‘8*X^3-12*X’ ,
And HERMITE(5) = ‘32*x^5-160*X^3+120*X’.
The HORNER function
The function HORNER produces the Horner division, or synthetic division, of a
polynomial P(X) by the factor (X-
a). The input to the function are the
polynomial P(X) and the number a. The function returns the quotient
polynomial Q(X) that results from dividing P(X) by (X- a), the value of a, and the
value of P( a), in that order. In other words, P(X) = Q(X)(X-a)+P(a). For
example, HORNER(‘X^3+2*X^2-3*X+1’,2) = {‘X^2+4*X+5’, 2, 11}. We
could, therefore, write X
3 +2X 2 -3X+1 = (X 2 +4X+5)(X-2)+11. A second
example: HORNER(‘X^6-1’,-5)= {’X^5-5*X^4+25*X^3-125*X^2+625*X-3125’,-5,
15624} i.e., X 6 -1 = (X 5 -5*X 4 +25X 3 -125X 2 +625X-3125)(X+5)+15624.
The variable VX
A variable called VX exists in the calculator’s {HOME CASDIR} directory that
takes, by default, the value of ‘X’. This is the name of the preferred
independent variable for algebraic and calculus applications. Avoid using
the variable VX in your programs or equations, so as to not get it confused
with the CAS’ VX. If you need to refer to the x-component of velocity, for
example, you can use vx or Vx. For additional information on the CAS
variable see Appendix C.