A comprehensive study of the generalized Lambert series
$\sum _{n=1}^{\infty }\frac{n^{N-2h}\text{exp}(-an^{N}x)}{1-\text{exp}(-n^{N}x)},0<a\leqslant 1,~x>0$
,
$N\in \mathbb{N}$
and
$h\in \mathbb{Z}$
, is undertaken. Several new transformations of this series are derived using a deep result on Raabe’s cosine transform that we obtain here. Three of these transformations lead to two-parameter generalizations of Ramanujan’s famous formula for
$\unicode[STIX]{x1D701}(2m+1)$
for
$m>0$
, the transformation formula for the logarithm of the Dedekind eta function and Wigert’s formula for
$\unicode[STIX]{x1D701}(1/N),N$
even. Numerous important special cases of our transformations are derived, for example, a result generalizing the modular relation between the Eisenstein series
$E_{2}(z)$
and
$E_{2}(-1/z)$
. An identity relating
$\unicode[STIX]{x1D701}(2N+1),\unicode[STIX]{x1D701}(4N+1),\ldots ,\unicode[STIX]{x1D701}(2Nm+1)$
is obtained for
$N$
odd and
$m\in \mathbb{N}$
. In particular, this gives a beautiful relation between
$\unicode[STIX]{x1D701}(3),\unicode[STIX]{x1D701}(5),\unicode[STIX]{x1D701}(7),\unicode[STIX]{x1D701}(9)$
and
$\unicode[STIX]{x1D701}(11)$
. New results involving infinite series of hyperbolic functions with
$n^{2}$
in their arguments, which are analogous to those of Ramanujan and Klusch, are obtained.