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3.2. Strategies for Factoring Polynomials over Z 89

E.30. A sequence of polynomials tin(t) is defined by the recurrence:

ulJ(t) = 0 q(t) = 1

uZL,(t) = %-l(t) - 21,-z (n 12).

Write out and factor the first few terms of the sequence. Look for any
interesting relationships among the polynomials in the sequence. Evaluate
the polynomials in the sequence at t = 1,2,3,4. Here are a number of facts
for you to verify:

(1)

(‘4 %I@> =

(3)

(4) u,(t) =

(5) u; - u;

U,(t)=t”-l- ( n;2)v+ (“pp-5

• n-4
  (^3 >

p-7 +...

(t2 - 4)-l/‘(y” - y-“) where y = +[t + (t2 - 4)‘/“]

un(i) = 2-+-1) [( ;)w+ (;)(t2-4)t”-3

+

( >

; (t2 - 4)2f3-5 +...

I

sin ntr

sin where t = 2~0~0. What are the zeros of un(t)?

= %a-k%+k (0 5 k 5 n)

(6) u:+l - 4 = uzn+l (0 5 n)

(7)^4 = un--lun+l +^1 (1 _< n)

(8) UI + ~3 +... + u-h-1 = ui (1 5 n)

(9) un + %+2k = (‘1lk+l - Uk-l)%+k (0 I n; 1 2 k)

(lo) ‘IIn + %+Zk+l = (uktl - uk)(%+k + ‘1ln+k+l) (0 5 k, n)

(11) U2k = (uk+l - uk-1)uk (1 5 k)

(12) UZk+l = (uk+l - uk)(uk+l + uk) (0 5 k)

(13) ‘11, = $[tu,-1 + ((t’ - 4)ui-, + 4)‘/“] (1 5 n)

(14) If m In, then u, I un (as polynomials with integer coefficients)