Abstract
In this paper, we give asymptotic expansions and inequalities related to the generalized Somos quadratic recurrence constant, using its relation with the generalized Euler constant.
Highlights
Formula (1.5) is closely related to Somos’ quadratic recurrence constant σ
Somos’ quadratic recurrence constant is defined by σ= √ 2 3···= ∞ n1/2n = 1 1/2k 1+ = exp k∞ ln k 2k n=1 k=1 k=1= 1.66168794 . . . (1.1)or σ = exp – dx = exp – dx dy
Chen and Han [7] pointed out that the lower bound in (1.14) is for n ≥ 24 sharper than the one in (1.12), and the upper bound in (1.14) is for n ≥ 18 sharper than the one in (1.12), For any positive integer m ≥ 2, in this paper we give the asymptotic expansion of γ (1/m) – γn(1/m) as n → ∞
Summary
Formula (1.5) is closely related to Somos’ quadratic recurrence constant σ . For n ∈ N, and presented the following asymptotic expansion: γ 2 – γn 2 These authors gave a formula for successively determining the coefficients in (1.15).
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