MA-410 Homework 1

Due at 4:59pm in my mailbox in SAS 3151, Tuesday, February 4, 2019



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-4 Polynomial Wn(x) in the variable x with integer coefficients, defined by the linear recurrence W0(x) = 1, W1(x) = 2x+1, Wn+2(x) = 2xWn+1(x) - Wn(x) for all n ≥ 0 one has
    (y-1/y) Wn( (y2 + 1/y2)/2 ) = y2n+1 - 1/y2n+1 for all n ≥ 0, real numbers y ≠ 0.
  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(e), page 25.
  4. Please compute integers x,y,z such that 21x + 33y + 77z = 1. Please show your work, computing a solution to 21x + 33y = 3 and 3w + 77z = 1 first.
  5. Prove for the extended Euclidean algorithm described in class: sk tk+1 - sk+1 tk = (-1)k+1 for all -1 ≤ k ≤ n-1.