Asymptotics and exact formulas for Zagier polynomials

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 $B_{n}^{*}$ and showed that they satisfy amusing variants of some properties of Bernoulli numbers. In particular, he studied the asymptotic behavior of $B_{2n}^{*}$, and also obtained an exact formula for them, the motivation for which came from the representation of $B_{2n}$ in terms of the Riemann zeta function $\zeta(2n)$. The modified Bernoulli numbers were recently generalized to Zagier polynomials $B_{n}^{*}(x)$. For $0<x<1$, an exact formula for $B_{2n}^{*}(x)$ 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 $B_{2n+1}^{*}(x)$ is also presented. The $6$-periodicity of $B_{2n+1}^{*}$ is deduced as a limiting case of it. These formulas are reminiscent of the Fourier expansions of Bernoulli polynomials. Some new results, for example, the one yielding the derivative of the Bessel function of the first kind with respect to its order as the Fourier coefficient of a function involving Chebyshev polynomials, are obtained in the course of proving these exact formulas. The asymptotic behavior of Zagier polynomials is also derived from them. Finally, a Zagier-type exact formula is obtained for $B_{2n}^{*}\left(-\frac{3}{2}\right)+B_{2n}^{*}$.