Chebyshev Polynomials: Properties of Derivatives

1. Show that:
2. $$T_n'(x)$$=$$2n\sum_{k=0\\k+n~~odd}^{n-1}\frac{1}{c_k}T_k(x)$$

3. $$T_n''(x)$$=$$\sum_{k=0\\k+n even}^{n-2}\frac{1}{c_k}n(n^2-k^2)T_k(x)$$ where $$c_0=2$$ and $$c_n=1$$ for $$n\geq1$$

I tried using the differential recursive relation of chebyshev polynomials but was unable to establish a relation that would help me solve the problem.

• For which values of n have you manually verified the equations? Sometime you will see a pattern or catch an error in the text. – Carl Christian Apr 24 at 11:59
• I was able to to use the recursive differential equation to show for the 1st order derivative. but i've not been able to identify one such method for the second one – Santa Apr 24 at 12:04