# Meeting Details

Title: Zagier polynomials and modified N\"{o}rlund polynomials Algebra and Number Theory Seminar Atul Dixit In 1998, Don Zagier studied the numbers B_{n}^{*} which he called the '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 we have shown that a theory parallel to that of 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 a similar thing is true for the Zagier polynomials. This exact formula involves Chebyshev polynomials and infinite series of Bessel function $Y_{n}(z)$. Through a motivation coming from diffraction theory, C. M. Linton has already proved this, but in a disguised form, and our proof is new and gives new results along the way. We also derive Zagier's formula as a limiting case of this general formula, which is interesting in itself. In the second part of my talk, I will discuss another generalization of the modified Bernoulli numbers that we recently studied along with A. Kabza, namely 'modified N\"{o}rlund polynomials' B_{n}^{(\alpha)*}, \alpha\in\mathbb{N}, and obtain their generating function along with applications.​ The talk will include an interesting mix of special functions, number theory, probability and umbral calculus.