Some Remarks on the Mathieu Series

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. 1. Introduction The infinite series is called a Mathieu series. It was introduced in 1890 by é. L. Mathieu (1835–1890) who studied various problems in mathematical physics. Since its introduction the series has been studied intensively. Mathieu himself conjectured that . The conjecture was proved in 1952 by Berg in [1]. Nowadays, the mathematical literature provides a range of papers dealing with inequalities for the series . In 1957 Makai [2] derived the double inequality More recently, Alzer et al. proved in [3] that Here, as usual, denotes the Riemann zeta function defined by , . The constants and are the best possible. Other lower and upper bound estimates for the Mathieu series can be found in the articles of Qi et al. [4] and Hoorfar and Qi [5]. An integral representation for the Mathieu series (1) is given by The integral representation was used by Elbert in [6] to derive the asymptotic expansion of : where denote the even indexed Bernoulli numbers defined by the generating function See also [7] for a derivation. The Mathieu series admits various generalizations that have been introduced and investigated intensively in recent years. The generalizations include the alternating Mathieu series, the -fold generalized Mathieu series, Mathieu -series, and Mathieu -series [8–11]. The generalizations recapture the classical Mathieu series as a special case. On the other hand, the alternating Mathieu series, although connected to its classical companion, is a variant that allows a separate study. It was introduced by Pogány et al. in [11] by the equation It possesses the integral representation Recently derived bounding inequalities for alternating Mathieu-type series can be found in the paper of Pogány and Tomovski [12]. The latest research article on integral forms for Mathieu-type series is the paper of Milovanovi？ and Pogány [13]. The authors present a systematic treatment of the subject based on contour integration. Among others, the following new integral representation for is derived [13, Corollary 2.2]: In this note we restrict the