Abstract
We present two classes of asymptotic expansions related to Somos’ quadratic recurrence constant and provide the recursive relations for determining the coefficients of each class of the asymptotic expansions by using Bell polynomials and other techniques. We also present continued fraction approximations related to Somos’ quadratic recurrence constant.
Highlights
Somos [1] defined the sequence g0 = 1, gn = ngn2–1, n ∈ N := {1, 2, 3, . . .}.The first few terms are g0 = 1, g1 = 1, g2 = 2, g3 = 12, g4 = 576, g5 = 1,658,880, . . . .The following asymptotic expansion is known in the literature: gn ∼ σ 2n n + 2 n 1 n2 4 n3 21 n4 138 n5
Nemes [15] studied the coefficients in the asymptotic expansion (1.1) and developed recurrence relations
Our last aim in this paper is to present continued fraction approximations related to Somos’ quadratic recurrence constant (Theorems 3.1 and 3.2)
Summary
Nemes [15] studied the coefficients in the asymptotic expansion (1.1) and developed recurrence relations. Where the coefficients ak (for k ∈ N0 := N ∪ {0}) are given by the recurrence relation a0 = 1, a1 = 2, a2 = –1, k–1 ak = The coefficients ak satisfy the following recurrence relation [15, Theorem 3]: a0 = 1, 1 ak = k k Where bk are the ordered Bell numbers defined by the exponential generating function [18, p.
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