Asymptotics and exact formulas for Zagier polynomials
Atul Dixit, M. Lawrence Glasser, Victor H. Moll, Christophe Vignat

TL;DR
This paper derives exact formulas and asymptotic behaviors for Zagier polynomials, involving Bessel functions and Chebyshev polynomials, extending Zagier's original work on modified Bernoulli numbers.
Contribution
It provides new explicit formulas for Zagier polynomials, including series involving Bessel functions and Chebyshev polynomials, and explores their asymptotic properties and periodicity.
Findings
Exact formulas involving Bessel and Chebyshev functions for Zagier polynomials.
Asymptotic behavior of Zagier polynomials derived from these formulas.
Identification of 6-periodicity in modified Bernoulli numbers as a limiting case.
Abstract
In 1998 Don Zagier introduced the modified Bernoulli numbers and showed that they satisfy amusing variants of some properties of Bernoulli numbers. In particular, he studied the asymptotic behavior of , and also obtained an exact formula for them, the motivation for which came from the representation of in terms of the Riemann zeta function . The modified Bernoulli numbers were recently generalized to Zagier polynomials . For , an exact formula for involving infinite series of Bessel function of the second kind and Chebyshev polynomials, that yields Zagier's formula in a limiting case, is established here. Such series arise in diffraction theory. An analogous formula for is also presented. The -periodicity of is deduced as a limiting case of it. These formulas are…
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Taxonomy
TopicsAdvanced Mathematical Identities · Mathematical functions and polynomials · Advanced Combinatorial Mathematics
