Some asymptotic expansions on hyperfactorial functions and generalized Glaisher–Kinkelin constants
Tóm tắt
In this paper, by the Bernoulli numbers and the exponential complete Bell polynomials, we establish two general asymptotic expansions on the hyperfactorial functions
$$\prod _{k=1}^nk^{k^q}$$
and the generalized Glaisher–Kinkelin constants
$$A_q$$
, where the coefficient sequences in the expansions can be determined by recurrences. Moreover, the explicit expressions of the coefficient sequences are presented and some special asymptotic expansions are discussed. It can be found that some well-known or recently published asymptotic expansions on the factorial function n!, the classical hyperfactorial function
$$\prod _{k=1}^nk^k$$
, and the classical Glaisher–Kinkelin constant
$$A_1$$
are special cases of our results, so that we give a unified approach to dealing with such asymptotic expansions.
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