ASYMPTOTIC-BEHAVIOR OF SOLUTIONS OF POINCARE DIFFERENCE-EQUATIONS

It is shown that if the zeros lambda1, lambda2, . . . , lambda(n) of the polynomial q(lambda) = lambda(n) + a1lambda(n-1) + . . . + a(n) are distinct and r is an integer in {1, 2, . . . , n} such that \lambda(s)\ not-equal \lambda(r)\ if s not-equal r, then the Poincare difference equation y(n + m) + (a1 + p1(m))y(n + m - 1) + . . . + (a(n) = p(n)(m))y(m) = 0 has a solution y(r) such that (A) y(r)(m) = lambda(r)m(1 + o(1)) as m --> infinity, provided that the sums SIGMA(j=m)infinity p(i)(j) (1 less-than-or-equal-to i less-than-or-equal-to n) converge sufficiently rapidly. Our results improve over previous results in that these series may converge conditionally, and we give sharper estimates of the o(1) terms in (A).