Modified Nörlund polynomials

Abstract

The modified Bernoulli numbers $$B_{n}^{*}$$ considered by Zagier are generalized to modified Norlund 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.