Abstract
By using the theory of the elliptic integrals, a new method of summation is proposed for a certain class of series and their derivatives involving hyperbolic functions. It is based on the termwise differentiation of the series with respect to the elliptic modulus and integral representations of several of the series in terms of the inverse Mellin transforms related to the Riemann zeta function. The relation with the corresponding case of the Voronoi summation formula is exhibited. The involved series are expressed in closed form in terms of complete elliptic integrals of the first and second kind, and some special cases are calculated in terms of particular values of the Euler gamma function.
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Acknowledgments
The author is sincerely indebted to the referee for pointing out the pioneer references on the topic and his comment, which led to the idea to extend the method for series involving the hyperbolic tangent and cotangent functions.
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The work was partially supported by CMUP (UID/MAT/00144/2013), which is funded by FCT(Portugal) with national (MEC) and European structural funds through the programs FEDER, under the partnership agreement PT2020.
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Yakubovich, S. On the curious series related to the elliptic integrals. Ramanujan J 45, 797–815 (2018). https://doi.org/10.1007/s11139-016-9861-6
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DOI: https://doi.org/10.1007/s11139-016-9861-6
Keywords
- Series with hyperbolic functions
- Elliptic integrals
- Mellin transform
- Riemann zeta function
- Euler gamma function
- Arithmetic functions
- Summation formulae