  We discuss a class of Little String Theories (LSTs) whose low energy
descriptions are supersymmetric gauge theories on the $\Omega$-background with
gauge group $U(N)$ and matter in the adjoint representation. We show that the
instanton partition function of these theories can be written in terms of
Kronecker-Eisenstein series, which in a particular limit of the deformation
parameters of the $\Omega$-background organise themselves into Greens functions
of free scalar fields on a torus. We provide a concrete identification between
(differences of) such propagators and Nekrasov subfunctions. The latter are
also characterised by counting specific holomorphic curves in a Calabi-Yau
threefold $X_{N,1}$ which engineers the LST. Furthermore, using the formulation
of the partition function in terms of the Kronecker-Eisenstein series, we argue
for new recursive structures which relate higher instanton contributions to
products of lower ones.
