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

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.

9 pages

References in corpus (3)

• Eight interesting identities involving the exponential function, derivatives, and Stirling numbers of the second kind
• Limit formulas for ratios of derivatives of the gamma and digamma functions at their singularities
• Limit formulas for ratios of polygamma functions at their singularities

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