Explicit formulas for computing Bernoulli numbers of the second kind and Stirling numbers of the first kind

TL;DR

This paper introduces explicit formulas for Bernoulli numbers of the second kind and Stirling numbers of the first kind, derived from a new formula for derivatives of the reciprocal logarithm, leading to recursive relations and identities.

Contribution

The paper provides novel explicit formulas and recursive relations for Bernoulli numbers of the second kind and Stirling numbers of the first kind, along with new identities and integral representations.

Findings

New explicit formulas for Bernoulli numbers of the second kind.

Recursive formulas for Stirling numbers of the first kind.

Derived identities and integral representations related to Stirling numbers.

Abstract

In the paper, by establishing a new and explicit formula for computing the $n$-th derivative of the reciprocal of the logarithmic function, the author presents new and explicit formulas for calculating Bernoulli numbers of the second kind and Stirling numbers of the first kind. As consequences of these formulas, a recursion for Stirling numbers of the first kind and a new representation of the reciprocal of the factorial $n!$ are derived. Finally, the author finds several identities and integral representations relating to Stirling numbers of the first kind.