Malmsten integrals, central factorial numbers, and the Eulerian polynomials

Overview & Synopsis

In this paper, we study certain classes of nested logarithmic integrals called Malmsten integrals and generalize certain results obtained by Blagouchine. These nested logarithmic integrals yield solutions in terms of sums of odd Riemann zeta and even Dirichlet beta values. We establish that the coefficients of these sums involve the central factorial numbers. Moreover, we provide an alternative proof for the expression of single instances of these values by means of the Eulerian polynomials and their midpoint variant. Finally, we derive consequences of these identities by making use of their self reciprocal properties. We thus establish a novel representation for the Midpoint Eulerian polynomials.