Chebyshev Or Tchebycheff Polynomials - HP 50g User Manual

The modified Bessel functions of the second kind,
are also solutions of this ODE.
You can implement functions representing Bessel's functions in the calculator in
a similar manner to that used to define Bessel's functions of the first kind, but
keeping in mind that the infinite series in the calculator need to be translated
into a finite series.

Chebyshev or Tchebycheff polynomials

The functions T
n
n = 0, 1, ... are called Chebyshev or Tchebycheff polynomials of the first and
second kind, respectively. The polynomials Tn(x) are solutions of the differential
2
equation (1-x ) (d
In the calculator the function TCHEBYCHEFF generates the Chebyshev or
Tchebycheff polynomial of the first kind of order n, given a value of n > 0. If
the integer n is negative (n < 0), the function TCHEBYCHEFF generates a
Tchebycheff polynomial of the second kind of order n whose definition is
You can access the function TCHEBYCHEFF through the command catalog
(‚N).
The first four Chebyshev or Tchebycheff polynomials of the first and second kind
are obtained as follows:
0 TCHEBYCHEFF, result: 1,
-0 TCHEBYCHEFF, result: 1,
1 TCHEBYCHEFF, result: 'X',
-1 TCHEBYCHEFF, result: 1,
2 TCHEBYCHEFF, result: '2*X^2-1,
-2 TCHEBYCHEFF, result: '2*X',
3 TCHEBYCHEFF, result: '4*X^3-3*X',
-3 TCHEBYCHEFF, result: '4*X^2-1',
K (x) = ( /2) [I
-1
(x) = cos(n cos
2 2
y/dx ) x (dy/dx) + n
U (x) = sin(n arccos(x))/sin(arccos(x)).
n
(x) I (x)]/sin
-
x), and U (x) = sin[(n+1) cos
n
2
y = 0.
i.e.,
i.e.,
i.e.,
i.e.,
i.e.,
i.e.,
i.e.,
i.e.,
,
-1
T (x) = 1.0.
0
U (x) = 1.0.
0
T (x) = x.
1
U (x) =1.0.
1
2
T (x) =2x -1.
2
U (x) =2x.
2
3
T (x) = 4x -3x.
3
2
U (x) = 4x -1.
3
2 1/2
x]/(1-x ) ,
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