Zagier polynomials and modified Norlund polynomials

Abstract : In 1998, Don Zagier studied the 'modified Bernoulli numbers' $B_{n}^{*}$ whose 6-periodicity for odd n naturally arose from his new proof of the Eichler-Selberg trace formula. These numbers satisfy amusing variants of the properties of the ordinary Bernoulli numbers. Recently, Victor H. Moll, Christophe Vignat and I studied an obvious generalization of the modified Bernoulli numbers, which we call 'Zagier polynomials'. These polynomials are also rich in structure, and we have shown that a theory parallel to that of the ordinary Bernoulli polynomials exists. Zagier showed that his asymptotic formula for $B_{2n}^{*}$ can be replaced by an exact formula. In an ongoing joint work with M. L. Glasser and K. Mahlburg, we have shown that an exact formula exists for the Zagier polynomials too. It involves Chebyshev polynomials and an infinite series of Bessel function Y_{n}(z). We also derive Zagier's exact formula as a limiting case of this general formula, which is interesting in itself. In the second part of my talk, we will discuss another generalization of the modified Bernoulli numbers that we recently studied along with A. Kabza, namely the modified Norlund polynomials $B_{n}^{(\alpha)*}$ for a natural number $\alpha$, and obtain their generating function along with applications.? The talk will include an interesting mix of special functions, number theory, probability and umbral calculus.