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Tuesday, October 21, 2014

**Abstract:** In 1998, Don Zagier studied the numbers $B_{n}^{*}$ which he called 'modified Bernoulli numbers'. They 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 a theory parallel to that of ordinary Bernoulli polynomials exists. One thing that was missing was a generalization of Zagier's beautiful exact formula for $B_{2n}^{*}$ for the Zagier polynomials. In an ongoing joint work with M. L. Glasser and K. Mahlburg, we have been able to obtain this generalization which involves Chebyshev polynomials and infinite series of Bessel function $Y_{n}$. I will mainly focus on these results. In the second part of my talk, I will discuss another generalization of the modified Bernoulli numbers that we studied along with A. Kabza, namely 'modified Nörlund polynomials' $B_{n}^{(\alpha)*}$ , $\alpha\in\mathbb{N}$, and obtain their generating function along with applications.