Some new identities and inequalities for Bernoulli polynomials and numbers of higher order related to the Stirling and Catalan numbers

The aim of this paper is to give some new relations, identities, and inequalities for the Bernoulli polynomials and numbers of higher order, the Stirling numbers of the second kind, the Eulerian numbers, and the Catalan numbers. By applying the Laplace transformation to the generating function of the Bernoulli polynomials of higher order, a novel formula for these polynomials is obtained. Integral and series representations for these polynomials and numbers are given. Moreover, the upper bound and the lower bound for the Bernoulli numbers of negative order are given. Some inequalities including the Bernoulli numbers of negative order and the Stirling numbers of the second kind are also given. Finally, appropriate ligaments of the definitions and results introduced here with those in earlier as well as oncoming investigations will be designated.