Chebyshev polynomials U_{n} form a series of orthogonal
polynomials. Compare your result with the function U(n,x)
given in the package ``orthopoly''.
Draw the graphs of U_{1}, U_{2}, U_{3},...,U_{6}.
Use the command ``combine'' to express sin^{k}x as a linear
combination of 2^{n-1}sin(nx),k = 1,3,5,7,9.
Let A be a 2×2 matrix. Suppose that A has eigenvalues a
and b. It is known that the determinant of A is
m
= ab and the trace of A is t = a+b.
Since a^{n} and b^{n} are the eigenvalues
of A^{n}, it follows that the determinant of A^{n}
is a^{n}b^{n} while the trace of A^{n}
is T_{n} = a^{n}+b^{n}.
Find the recurrence relations satisfied by T_{n} for n
³
0.
Express T_{n} as a function of t and m for
n
= 0,1,2,...,20.
Find the pattern of
for n = 1,2,3,···. Use the spreadsheet to find
its expansion.
The shifted Chebyshev polynomials are given by
T_{k}^{*}(x)
= T_{k}(2x-1)
Find the recursive relation satisfied by T_{k}^{*}.
Compute the coefficients of T_{k}^{*}
for k = 1, 2, ···, 10.
Express 2^{2n-1}x^{n} as a linear combination
of T_{k}^{*} for n
= 1,2,···,12.
Let the polynomials C_{n}(x) be given recursively
by
C_{0}(x) = 2 , C_{1}(x)
= x , C_{n}_{+1}(x) = xC_{n}(x)-C_{n}_{-1}(x)
for
n ³ 1.
List the coefficients of C_{n}(x) for n =
0,1,2,···,12.
Express x^{n} as a linear combination of C_{k}(x)
for n = 0,1,2,···,12.
Let the polynomials S_{n}(x) be given recursively
by
S_{0}(x) = 1 , S_{1}(x)
= x , S_{n}_{+1}(x) = xS_{n}(x)-S_{n}_{-1}(x)
for
n ³ 1.
List the coefficients of S_{n}(x) for n =
0,1,2,···,12.
Express x^{n} as a linear combination of S_{k}(x)
for n = 0,1,2,···,12.