On asymptotic constants related to products of Bernoulli numbers and factorials

Abstract

We discuss the asymptotic expansions of certain products of Bernoulli numbers and factorials, e.g., \[ \prod_{ν=1}^n |B_{2ν}| \quad \text{and} \quad \prod_{ν=1}^n (k ν)!^{ν^r} \quad \text{as} \quad n \to \infty \] for integers $k \geq 1$ and $r \geq 0$. Our main interest is to determine exact expressions, in terms of known constants, for the asymptotic constants of these expansions and to show some relations among them.