MA410 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.

Prove by induction that for the nth Chebyshev2 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)));
 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(c), page 25.

Please compute integers x,y,z such that 24x + 44y + 33z = 1.
Hint: compute a solution 24x + 44y = 4 and 4w + 33z = 1.

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