MA410 Homework 1
Due at 4:59pm in my mailbox in SAS 3151, Tuesday, February 5, 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.

Prove by induction that for the nth Chebyshev3 Polynomial V_{n}(x) in the variable x
with integer coefficients,
defined by the linear recurrence V_{0}(x) = 1, V_{1}(x) = 2x1,
V_{n+2}(x) = 2xV_{n+1}(x)  V_{n}(x) for all n ≥ 0 one has
(y+1/y) V_{n}( (y^{2} + 1/y^{2})/2 )
= y^{2n+1} + 1/y^{2n+1} for all n ≥ 0, real numbers y ≠ 0.

Consider C_{n} from ENT, §1.2, Problem 10, page 12.
Prove that C_{n} = binomial(2n,n)  binomial(2n,n1) for
all n ≥ 1, thus showing that the C_{n} are integers.

ENT, §2.2, Problem 20(f), page 25.
Prove first that GCD(a,b^{2}) = 1,
and then GCD(A^2, B) = 1 for A=a and B = b^2.

Please compute integers x,y,z such that 20x + 28y + 35z = 1.
Please show your work, computing a solution to 20x + 28y = 4 and
4w + 35z = 1 first.

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