  The aim of this paper is to study certain multiple series which can be
regarded as multiple analogues of Eisenstein series. As a prior research, the
second-named author considered double analogues of Eisenstein series and
expressed them as polynomials in terms of ordinary Eisenstein series. This fact
was derived from the analytic observation of infinite series involving
hyperbolic functions which were based on the study of Cauchy, and also
Ramanujan. In this paper, we prove an explicit relation formula among these
series. This gives an alternative proof of this fact by using the technique of
partial fraction decompositions of multiple series which was introduced by
Gangl, Kaneko and Zagier. By the same method, we further show a certain
multiple analogue of this fact and give some examples of explicit formulas.
Finally we give several remarks about the relation between our present result
and the previous work for infinite series involving hyperbolic functions.
