Modified Nörlund Polynomials

Abstract : The modified Bernoulli numbers B_{n}^{*} considered by Zagier are generalized to modified Nörlund 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.