Functional relations for hyperbolic cosecant series

2021/02/17 by Buzzegoli, M.
#11B68 (Secondary) #11L03 (Primary) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2102.08676

Abstract

We study the function series ∑_n=1^∞ ϕ^2m+2 cosch^2m+2(nϕ/2), and similar series, for integers m and complex ϕ. 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 ϕ=± 2πi. We furthermore derive several identities between Harmonic numbers and ordinary and generalized Bernoulli polynomials.

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