Explicit formulas for degenerate Bernoulli numbers

Abstract The ‘degenerate’ Bernoulli numbers β m ( 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 β m ( λ ) is a polynomial in λ of degree ⩽ m . In this paper we find explicit formulas for the coefficients of the polynomial β m ( λ ), and we give a new proof of the degenerate Staudt-Clausen theorem. New recursion formulas for β m ( λ ) are also proved.