Functional relations for hyperbolic cosecant series

Abstract

We study the function series ${\sum }_{n=1}^{\infty }{\varphi }^{2m+2}{\text{cosch}}^{2m+2}(n\varphi /2)$, and similar series, for integers $m$ and complex $\varphi$. This hyperbolic series is linearly related to the Lambert series. The Lambert series is known to satisfy a functional equation which defines the Ramanujan polynomials. By using residue theorem (summation theorem) we find the functional equation satisfied by this hyperbolic series. The functional equation identifies a class of polynomials which can be seen as a generalization of the Ramanujan polynomials. These polynomials coincide with the asymptotic expansion of the hyperbolic series at the origin and they all vanish for $\varphi =\pm 2\pi i$. We furthermore derive several identities between Harmonic numbers and ordinary and generalized Bernoulli polynomials.

34 pages, 3 figures

References in corpus (1)

• The Unruh effect and its applications