MA410 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.

Prove by induction that for the nth Chebyshev4 Polynomial W_{n}(x) in the variable x
with integer coefficients,
defined by the linear recurrence W_{0}(x) = 1, W_{1}(x) = 2x+1,
W_{n+2}(x) = 2xW_{n+1}(x)  W_{n}(x) for all n ≥ 0 one has
(y1/y) W_{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(e), page 25.

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.

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 ≤ n1.