For more information about this meeting, contact Robert Vaughan, Karl Schwede, Mihran Papikian, Ae Ja Yee.

Title: | Zagier polynomials and modified N\"{o}rlund polynomials |

Seminar: | Algebra and Number Theory Seminar |

Speaker: | Atul Dixit |

Abstract: |

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. |

### Room Reservation Information

Room Number: | MB106 |

Date: | 11 / 20 / 2014 |

Time: | 11:15am - 12:05pm |