Asymptotics and exact formulas for Zagier polynomials

Show simple item record Dixit, Atul Glasser, M. Lawrence Moll, Victor H. Vignat, Christophe 2016-07-19T06:17:23Z 2016-07-19T06:17:23Z 2016-12
dc.identifier.citation Dixit, Atul; Glasser, M. Lawrence; Moll, Victor H. and Vignat, Christophe, “Asymptotics and exact formulas for Zagier polynomials”, Research in Number Theory, DOI: 10.1007/s40993-016-0044-8, vol. 2, no. 1, Dec. 2016. en_US
dc.identifier.issn 2363-9555
dc.description.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 B2n in terms of the Riemann zeta function ζ(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(−32)+B∗2n en_US
dc.description.statementofresponsibility by Atul Dixit, M. Lawrence Glasser, Victor H. Moll and Christophe Vignat
dc.format.extent vol. 2, no. 1
dc.language.iso en_US en_US
dc.publisher Springer en_US
dc.subject Bernoulli numbers en_US
dc.subject Zagier polynomials en_US
dc.subject Bessel functions en_US
dc.subject Chebyshev polynomials en_US
dc.subject Fourier expansions en_US
dc.subject Differential equations en_US
dc.subject Asymptotics en_US
dc.subject Diffraction theory en_US
dc.title Asymptotics and exact formulas for Zagier polynomials en_US
dc.type Article en_US
dc.relation.journal Research in Number Theory

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