Bernoulli identities, zeta relations, determinant expressions, Mellin transforms, and representation of the Hurwitz numbers

JOURNAL OF NUMBER THEORY

卷:184

Article

Coffey, Mark W.¹ 

[1] Colorado Sch Mines, Dept Phys, Golden, CO 80401 USA

关键词: Bernoulli number;    Bernoulli polynomial;    Riemann zeta function;    Euler number;    Euler polynomial;    Alternating zeta function;    Double zeta values;    Hurwitz zeta function;    Lerch zeta function;    Polygamma function;    Ramanujan polynomial;    Bernoulli relations;    Zeta identities;    Eisenstein series;    Recurrence;   

DOI  :  10.1016/j.jnt.2017.08.009

来源: Elsevier

【 摘 要 】

The Riemann zeta identity at even integers of Lettington, along with his other Bernoulli and zeta relations, is generalized. Other corresponding recurrences and determinant relations are illustrated. Another consequence is the application to sums of double zeta values. A set of identities for the Ramanujan and generalized Ramanujan polynomials is presented. An alternative proof of Lettington's identity is provided, together with its generalizations to the Hurwitz and Lerch zeta functions, hence to Dirichlet L series, to Eisenstein series, and to general Mellin transforms. The Hurwitz numbers (H) over tilden occur in the Laurent expansion about the origin of a certain Weierstrass p function for a square lattice, and are highly analogous to the Bernoulli numbers. An integral representation of the Laurent coefficients about the origin for general p functions, and for these numbers in particular, is presented. As a Corollary, the asymptotic form of the Hurwitz numbers is determined. In addition, a series representation of the Hurwitz numbers is given, as well as a new recurrence. Other results concern the Matter numbers of the equianharmonic case of the p function. (C) 2017 Elsevier Inc. All rights reserved.

【 授权许可】

Free