  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
$B_{2n}^{*}\left(-\frac{3}{2}\right)+B_{2n}^{*}$.
