Abstract
The object of this note is to present new expressions for the classical Mathieu series in terms of hyperbolic functions. The derivation is based on elementary arguments concerning the integral representation of the series. The results are used afterwards to prove, among others, a new relationship between the Mathieu series and its alternating companion. A recursion formula for the Mathieu series is also presented. As a byproduct, some closed-form evaluations of integrals involving hyperbolic functions are inferred.
Highlights
The infinite series S (r) = ∞ ∑ n=1 (n2 2n + r2)2, r > 0, (1)is called a Mathieu series
The latest research article on integral forms for Mathieu-type series is the paper of Milovanovicand Pogany [13]
It is interesting to mention that (18) may be used to prove a result for a closed-form evaluation of an infinite series that is related to S(r)
Summary
The Mathieu series admits various generalizations that have been introduced and investigated intensively in recent years. It was introduced by Pogany et al in [11] by the equation. Derived bounding inequalities for alternating Mathieu-type series can be found in the paper of Pogany and Tomovski [12]. The latest research article on integral forms for Mathieu-type series is the paper of Milovanovicand Pogany [13]. The following new integral representation for S(r) is derived [13, Corollary 2.2]: π∫. Starting with the integral form (4) new representations for S(r) are derived. A new relationship between the Mathieu series and its alternating variant is derived. A new proof is given for an exact evaluation of an infinite series related to S(r)
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