Explicit formulas for degenerate Bernoulli numbers Original Research Article

The ‘degenerate’ Bernoulli numbers βm(λ) can be defined by means of the exponential generating function x((1 + λx)1/λ −1)−1. L. Carlitz proved an analogue of the Staudt-Clausen theorem for these numbers, and he showed that βm(λ) is a polynomial in λ of degree ⩽ m. In this paper we find explicit formulas for the coefficients of the polynomial βm(λ), and we give a new proof of the degenerate Staudt-Clausen theorem. New recursion formulas for βm(λ) are also proved.