Dixit, AtulAtulDixitGlasser, M. LawrenceM. LawrenceGlasserMoll, Victor H.Victor H.MollVignat, ChristopheChristopheVignat2025-08-302025-08-302016-12-0110.1007/s40993-016-0044-82-s2.0-85020015592https://d8.irins.org/handle/IITG2025/22607In 1998 Don Zagier introduced the modified Bernoulli numbers Bn∗ and showed that they satisfy amusing variants of some properties of Bernoulli numbers. In particular, he studied the asymptotic behavior of B2n∗, and also obtained an exact formula for them, the motivation for which came from the representation of B<inf>2</inf> <inf>n</inf> in terms of the Riemann zeta function ζ(2 n). The modified Bernoulli numbers were recently generalized to Zagier polynomials Bn∗(x). For 0 < x< 1 , an exact formula for B2n∗(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 B2n+1∗(x) is also presented. The 6-periodicity of B2n+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 B2n∗(-32)+B2n∗.trueAsymptotics | Bernoulli numbers | Bessel functions | Chebyshev polynomials | Differential equations | Diffraction theory | Fourier expansions | Zagier polynomialsArticle236395551 December 2016012arJournal0