Some identities for degenerate Bernoulli numbers of the second kind

TL;DR

This paper introduces degenerate Bernoulli numbers of the second kind, derives related differential equations, and provides explicit formulas and identities connecting these numbers to higher-order versions.

Contribution

It presents the first definition of degenerate Bernoulli numbers of the second kind and derives their differential equations and explicit formulas.

Findings

Derived nonlinear differential equations for the generating function

Obtained explicit formulas for degenerate Bernoulli numbers of the second kind

Established identities relating these numbers to higher-order versions

Abstract

We introduce the degenerate Bernoulli numbers of the second kind as a degenerate version of the Bernoulli numbers of the second kind. We derive a family of nonlinear differential equations satisfied by a function closely related to the generating function for those numbers. We obtain explicit expressions for the coefficients appearing in those differential equations and the degenerate Bernoulli numbers of the second kind. In addition, as an application and from those differential equations we have an identity expressing the degenerate Bernoulli numbers of the second kind in terms of those numbers of higher-orders.