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Explicit formulas for degenerate Bernoulli numbers. (English) Zbl 0873.11016

L. Carlitz [Arch. Math. 7, 28-33 (1956; Zbl 0070.04003)] introduced the ‘degenerate’ Bernoulli numbers ${\beta }_{m}\left(\lambda \right)$ by means of the generating function

$\frac{x}{{\left(1+\lambda x\right)}^{1/\lambda }-1}=\sum _{m=0}^{\infty }{\beta }_{m}\left(\lambda \right)\frac{{x}^{m}}{m!}·$

He also proved an analogue of the Staudt-Clausen theorem for these numbers, and he showed that ${\beta }_{m}\left(\lambda \right)$ is a polynomial in $\lambda$ of degree $\le m$.

In the paper under review the author gives explicit formulas for the coefficients of the polynomial ${\beta }_{m}\left(\lambda \right)$ and thereby an alternative proof of the Staudt-Clausen theorem, including some new recursion formulas for ${\beta }_{m}\left(\lambda \right)$.

##### MSC:
 11B68 Bernoulli and Euler numbers and polynomials