New asymptotic expansions on hyperfactorial functions

In this paper, by using the Bernoulli numbers and the exponential complete Bell polynomials, we establish four general asymptotic expansions for the hyperfactorial functions $\prod _{k=1}^n {k^{k^q}}$, which have only odd power terms or even power terms. We derive the recurrences for the parameter sequences in these four general expansions and give some special asymptotic expansions by these recurrences

New asymptotic expansions on hyperfactorial functions

In this paper, by using the Bernoulli numbers and the exponential complete Bell polynomials, we establish four general asymptotic expansions for the hyperfactorial functions $\prod _{k=1}^n {k^{k^q}}$, which have only odd power terms or even power terms. We derive the recurrences for the parameter sequences in these four general expansions and give some special asymptotic expansions by these recurrences

Khinchin Families and Hayman Class

We give criteria, following Hayman and Báez-Duarte, for non-vanishing functions with non-negative coefficients to be Gaussian and strongly Gaussian. We use these criteria to show in a simple and unified manner asymptotics for a number of combinatorial objects, and, particularly, for a variety of partition questions like Ingham’s theorem on partitions with parts in an arithmetic sequence, or Wright’s theorem on plane partitions and, of course, Hardy–Ramanujan’s partition theore

A generalization of Krull-Webster's theory to higher order convex functions: multiple Γ\Gamma-type functions

We provide uniqueness and existence results for the eventually $p$-convex and eventually $p$-concave solutions to the difference equation $\Delta f=g$ on the open half-line $(0,\infty)$, where $p$ is a given nonnegative integer and $g$ is a given function satisfying the asymptotic property that the sequence $n\mapsto\Delta^p g(n)$ converges to zero. These solutions, that we call $\log\Gamma_p$-type functions, include various special functions such as the polygamma functions, the logarithm of the Barnes $G$-function, and the Hurwitz zeta function. Our results generalize to any nonnegative integer $p$ the special case when $p=1$ obtained by Krull and Webster, who both generalized Bohr-Mollerup-Artin's characterization of the gamma function. We also follow and generalize Webster's approach and provide for $\log\Gamma_p$-type functions analogues of Euler's infinite product, Weierstrass' infinite product, Gauss' limit, Gauss' multiplication formula, Legendre's duplication formula, Euler's constant, Stirling's constant, Stirling's formula, Wallis's product formula, and Raabe's formula for the gamma function. We also introduce and discuss analogues of Binet's function, Burnside's formula, Fontana-Mascheroni's series, Euler's reflection formula, and Gauss' digamma theorem. Lastly, we apply our results to several special functions, including the Hurwitz zeta function and the generalized Stieltjes constants, and show through these examples how powerful is our theory to produce formulas and identities almost systematically