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Asymptotic expansions for certain mathematical constants and special functions

Chen Chao-Ping (Henan Polytechnic University, School of Mathematics and Informatics, Jiaozuo City, Henan Province, China)

For fixed real b > 1 and α > 0, let S[α]b (n) = Σn,k=1 bkk-α. Abel proved that S[α]b(n) ~ bn Σ∞,k =0 ckn-(k+α)(n → ∞), and gave an explicit formula for determining the coefficients ck ≡ ck(b,α) 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.

Keywords: Asymptotic expansion, Somos' quadratic recurrence constant, Glaisher-Kinkelin constant, Choi-Srivastava constants, Barnes G-function