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arXiv · 2102.08676

Functional relations for hyperbolic cosecant series

M. Buzzegoli
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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.

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M. Buzzegoli. 2021-02-17. Functional relations for hyperbolic cosecant series. https://arxiv.org/abs/2102.08676

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