MA-410 Homework 1

Due at 4:59pm in my mailbox in SAS 3151, Tuesday, February 3, 2015

All solutions must be submitted in hardcopy either to me in class or placed in my mailbox.
Note my office hours on my schedule.

1. Prove by induction that for the n-th Chebyshev Polynomial T_n(x) in the variable x, defined by the linear recurrence T_0 = 1, T_1(x) = x, T_{n+2}(x) = 2xT_{n+1}(x) - T_n(x) for all n ≥ 0 one has
   
   T_n( (y + 1/y)/2 ) = (y^n + y^(-n))/2 for all n ≥ 0, y ≠ 0.
   
2. Consider C_n from ENT, §1.2, Problem 10, page 12. Prove that C_n = binomial(2n,n) - binomial(2n,n-1) for all n ≥ 1, thus showing that the C_n are integers.
3. ENT, §2.2, Problem 20(f), page 25. Prove first that GCD(a,b²) = 1, and that for g = GCD(a², b²), GCD(a,g) = 1.
4. ENT, §2.4, Problem 2(b), page 31. Please compute the Bezout coefficients by the extended Euclidean algorithm presented in class, not the back-substitution given in the book.
5. Prove for the extended Euclidean algorithm described in class: s_k t_k+1 - s_k+1 t_k = (-1)^k+1 for all -1 ≤ k ≤ n