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5.3. Location of Complex Roots 185

In general, for fixed k, a recursion of the type

U,+l = UOV, + al&-1 + “. + akv,-k

can be written in the form

(v,+l + bovn +.. .) = r(v, + bov,-l +.. -)

provided r is a root of the equation rk+l = aOrk +... + ok. Knowing the
roots of this equation will provide us information on the general behavior of
the sequence. In the special case that all its roots rl, t-2,... , Pk are distinct,
it can be shown that

where the coefficients ci depend on vi, ~2,... , vk. Thus, if all ri are real
and less than 1 in absolute value, v, will become closer and closer to zero.
In any case, as n grows larger, lv,+Jv, I tends to the largest Iri I.
For each of the following recursions, write out the first 10 or so terms to
get some idea of how it behaves, whether increasing or decreasing without
bound, tending to zero, oscillating boundedly or unboundedly. In each case,
there is a certain polynomial whose zeros can be used to give a formula
for the general term of the recursion. Find the zeros of the polynomials
and consider how the behaviour of the sequence is related to the following
properties of the zeros:

(a) the zero of largest absolute value is positive and exceeds 1

(b) the zero of largest absolute value is negative and is less than 1

(c) the zero of largest absolute value lies outside the unit circle

(d) the zero of largest absolute value lies on the unit circle

(e) all zeros have absolute value less than 1

(f) all zeros are nth roots of unity for some integer n.

(i) u. = 1 u1 = 2 u,+i = 21, -21,-r (n 2 1)

(ii) uc = 2 ~1 = 3 21,+i = 3u, - 2u,-i (n > 1)

(iii) ~0 = 3 211 = 2 u,+i = 3u, -221,-l (n 1 1)

(iv) uc = ~1 = 1 621,+i = 721, - 2.1~“~~ (n 2 1)

(v) uo = ul = 212 = 1 21,+1 = 2~~ - u+i + 2~~~2 (n > 2)

(vi) us = ‘1~1 = us = 1 2u,+l = u, -226,-l + U1,-2 (n > 2)

(vii) us = ui = 1 212 = 2 zL,+i = U, - 2u,-i + 221,-z (n > 2).