Effective Asymptotics for Some Nonlinear Recurrences and Almost Doubly-exponential Sequences

We develop a technique to compute asymptotic expansions for recurrent sequences of the form an+1 = f(an), where f(x) = x − axα + bxβ + o(xβ) as x → 0, for some real numbers α, β, a, and b satisfying a > 0, 1 < α < β. We prove a result which summarizes the present stage of our investigation, generalizing the expansions in [Amer. Math Monthly, Problem E 3034[1984, 58], Solution [1986, 739]]. One can apply our technique, for instance, to obtain the formula: an = √ 3 √ n − 3 √ 3 10 lnn n √ n + 9 √ 3 50 lnn n2 √ n + o ( lnn n5/2 ) , where an+1 = sin(an), a1 ∈ IR. Moreover, we consider the recurrences an+1 = an + gn, and we prove that under some technical assumptions, an is almost doubly-exponential, namely an = bk2 nc, an = bk2 nc + 1, an = bk2 n − 1 2 c, or an = bk2 n + 5 2 c for some real number k, generalizing a result of Aho and Sloane [Fibonacci Quart. 11 (1973), 429–437].