A note on $lanbda$-Bernoulli numbers of the second kind

Abstract

In this paper, we study the λ -Bernoulli numbers of the second kind which are defined as an integral of the λ -analogue of the falling factorial sequence and express them in terms of the λ -Stirling numbers of the first kind. Then we investigate the generalized λ -Bernoulli numbers of the second kind given as a multiple integral on the unit cube and show, among other things, the generating function of those numbers can be expressed in term of the recently introduced poly- exponential function by Kim-Kim. Finally, we introduce the higher-order λ -Bernoulli numbers of the second kind, again given by another multiple integral on the unit cube, and show those numbers can be given by the degenerate Stirling numbers of the second.