The first few Chebyshev Polynomials of the first kind

For your convenience, here are some of the first few:

$$T_0(k) = 1$$

$$T_1(k) = x$$

$$T_2(k) = 2x^2-1$$

$$T_3(k) = 4x^3-3x$$

$$T_4(k) = 8x^4-8x^2 + 1$$

$$T_5(k) = 16x^5 - 20x^3 + 5x$$

The rest may be generated in Matlab using the following:

T(:, 1) = ones(length(x), 1);
T(:, 2) = x;
for n = 3:Hmax+1
  T(:, n) = 2*x.*T(:,n-1) - T(:,n-2);
end