  General summation formulas have been proved to be very useful in analysis,
number theory and other branches of mathematics. The Lipschitz summation
formula is one of them. In this paper, we give its application by providing a
new transformation formula which generalizes that of Ramanujan. Ramanujan's
result, in turn, is a generalization of the modular transformation of
Eisenstein series $E_k(z)$ on SL$_2(\mathbb{Z})$, where $z\to-1/z,
z\in\mathbb{H}$. The proof of our result involves delicate analysis containing
Cauchy Principal Value integrals. A simpler proof of a recent result of ours
with Kesarwani giving a non-modular transformation for
$\sum_{n=1}^{\infty}\sigma_{2m}(n)e^{-ny}$ is also derived using the Lipschitz
summation formula. In the pursuit of obtaining this transformation, we
naturally encounter a new generalization of Raabe's cosine transform whose
several properties are also demonstrated. As a corollary of this result, we get
a generalization of Wright's asymptotic estimate for the generating function of
the number of plane partitions of a positive integer $n$.
