Modified Nörlund polynomials

Abstract

The modified Bernoulli numbers $B_n^{\ast }$ considered by Zagier are generalized to modified Nörlund polynomials $B_n^{(\ell )\ast }$. For $\ell \in \mathbb{N}$, an explicit expression for the generating function for these polynomials is obtained. Evaluations of some spectacular integrals involving Chebyshev polynomials, and of a finite sum involving integrals of the Hurwitz zeta function are also obtained. New results about the $\ell$-fold convolution of the square hyperbolic secant distribution are obtained, such as a differential-difference equation satisfied by a logarithmic moment and a closed-form expression in terms of the Barnes zeta function.

submitted for publication