A generating function and a convolution for the Bernoulli numbers arising from finite-part integration

Abstract

Using finite-part integration, we uncover a generating function and a convolution of positive even Bernoulli numbers arising from the finite-part of the divergent integrals ∫₀^∞ csch t dt and ∫₀^∞ csch² t dt. This is done by utilizing the splitting property of the finite-part integral and through its Mellin transform representation. Other convolutions from applications of the results in this paper are then presented.