Modified N\"orlund polynomials

TL;DR

This paper introduces modified N"orlund polynomials, providing explicit generating functions, evaluating complex integrals, and exploring convolution properties related to hyperbolic secant distributions and special functions.

Contribution

It generalizes Zagier's modified Bernoulli numbers to N"orlund polynomials, deriving explicit formulas, integral evaluations, and convolution properties with special functions.

Findings

Explicit generating functions for modified N"orlund polynomials.

Evaluations of integrals involving Chebyshev polynomials and Hurwitz zeta function.

New convolution results for the square hyperbolic secant distribution.

Abstract

The modified Bernoulli numbers $B_{n}^{*}$ considered by Zagier are generalized to modified N\"orlund polynomials ${B_{n}^{(\ell)*}}$. 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.