On generalized Mathieu series and its companions

Abstract

Integral representations for a generalized Mathieu series and its companions are used to obtain bounds for their corresponding series. The bounds are procured mainly using results pertaining to theČebyšev functional. The relationship to Zeta type functions are also examined. It is demonstrated that the Zeta companion relations are a particular case of the generalised Mathieu companions.

On Zeta and Dirichlet Beta Function Families as Generators of Generalized Mathieu Series, Providing Approximation and Bounds

“…Theorem 2.1. (see [16] for the proof ).For µ > 0 and r > 0, the alternating generalized Mathieu series Sµ (r) satisfies the following bounds,…”

“…Using the generalized Mathieu series, S µ (r) as given in (1.1) and (1.6)-(1.7) together with the alternating generalized Mathieu series Sµ (r) as given in (1.11)-(1.12) we introduce the odd generalized Mathieu series, φ µ (r) and the even generalized Mathieu series, ψ µ (r). These are given by [16] (2.9)…”

“…In ( [16] )the odd and even generalized Mathieu series bounds were obtained via a Čebyšev functional approach. If we allow the subscripts of O and E to represent the cases related to φ µ (r) (odd) and ψ µ (r) (even).…”

“…In paper [16], bounds were obtained for the alternating generalized Mathieu series Sµ (r), the odd φ µ (r) and even ψ µ (r) generalized Mathieu series. This was accomplished by using their integral representations via Čebyšev Functional bounds which is presented in Subsection 2.1.…”

“…In the paper [11] the emphasis was to extend the methodology for the Zeta family to the Dirichlet Beta L-function family through the generalized Mathieu series which is presented in section 2. The work in [16] The generalized Mathieu series are based on two parameters r and µ, as exemplified by (1.5) and (1.6) in addition to various generators.The Dirichlet L-series have played a great deal of attention in number theory. These are also relevant to lattice sums which may be represented by lower dimensional lattice sums.The classic example of this was first given by Lorenz [26] as well in [39] , given by,…”