## MA-410 Homework 1 Due at 4:59pm in my mailbox in SAS 3151, Tuesday, February 6, 2018

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-2 Polynomial U_n(x) in the variable x, defined by the linear recurrence U_0 = 1, U_1(x) = 2x, U_{n+2}(x) = 2xU_{n+1}(x) - U_n(x) for all n ≥ 0 one has

U_n( (y + 1/y)/2 ) = 1/y^n (1 + y^2 + y^4 + ... + y^(2n) ), for all n ≥ 0, y ≠ 0.

You can check in Maple for n=11: expand(y^11*simplify(ChebyshevU(11,(y+1/y)/2)));
2. Consider Cn from ENT, §1.2, Problem 10, page 12. Prove that Cn = binomial(2n,n) - binomial(2n,n-1) for all n ≥ 1, thus showing that the Cn are integers.
3. ENT, §2.2, Problem 20(c), page 25.
4. Please compute integers x,y,z such that 24x + 44y + 33z = 1. Hint: compute a solution 24x + 44y = 4 and 4w + 33z = 1.
5. Prove for the extended Euclidean algorithm described in class: sk tk+1 - sk+1 tk = (-1)k+1 for all -1 ≤ k ≤ n