Abstract
For fixed real b > 1 and ? > 0, let S[?]b (n) = ?n,k=1 bkk-?. Abel proved that S[?]b(n) ~ bn ??,k =0 ckn-(k+?)(n ? ?), and gave an explicit formula for determining the coefficients ck ? ck(b,?) in terms of Stirling numbers of the second kind. We here provide a recurrence relation for determining the coefficients ck, without Stirling numbers. We also consider asymptotic expansions concerning Somos' quadratic recurrence constant, Glaisher-Kinkelin constant, Choi-Srivastava constants, and the Barnes G-function.
Highlights
Abel [1] derived a complete asymptotic expansion for a sequence of the following sum (1) Sb[α](n) =n bk kα k=1 as n → ∞, for fixed real b > 1 and α > 0
Stirling numbers of the second kind can be computed by the formula
We aim to provide a recurrence relation for determining the coefficients of n−(α+k) in Abel’s expansion (2), without help of Stirling numbers of the second kind
Summary
Abel [1] derived a complete asymptotic expansion for a sequence of the following sum (1). Asymptotic expansion, Somos’ quadratic recurrence constant, Glaisher-Kinkelin constant, Choi-Srivastava constants, Barnes G-function. Alzer et al [3] applied a classical series identity involving the psi function with a view to deriving series representations for a number of known mathematical constants. Chen and Srivastava [19] established new analytical representations for the Euler-Mascheroni constant in terms of the psi function. Chen and Srivastava [20] established several further analytical representations for the Euler-Mascheroni constant in terms of the psi function. We consider asymptotic expansions concerning Somos’ quadratic recurrence constant, Glaisher-Kinkelin constant, Choi-Srivastava constants, and the Barnes G-function.
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