Functional relations for hyperbolic cosecant series

TL;DR

This paper derives functional equations for hyperbolic cosecant series, linking them to Lambert series and Ramanujan polynomials, and uncovers new identities involving harmonic numbers and Bernoulli polynomials.

Contribution

It introduces a new class of polynomials generalizing Ramanujan polynomials through functional equations of hyperbolic series.

Findings

Functional equations for hyperbolic cosecant series derived

Identification of polynomial class generalizing Ramanujan polynomials

New identities between harmonic numbers and Bernoulli polynomials established

Abstract

We study the function series $\sum_{n=1}^\infty \phi^{2m+2} \text{cosch}^{2m+2}(n\phi/2)$, and similar series, for integers $m$ and complex $\phi$. 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 $\phi=\pm 2\pi i$. We furthermore derive several identities between Harmonic numbers and ordinary and generalized Bernoulli polynomials.