  We revisit, with a pedagogical heuristic motivation, the lambda extension of
the low-temperature row correlation functions C(M,N) of the two-dimensional
Ising model. In particular, using these one-parameter series to understand the
deformation theory around selected values of $\lambda$, namely $\lambda =
\cos(\pi \, m/n)$ with m and n integers, we show that these series yield
perturbation coefficients, generalizing form factors, that are D-finite
functions. As a by-product these exact results provide an infinite number of
highly non-trivial identities on the complete elliptic integrals of the first
and second kind. These results underline the fundamental role of Jacobi theta
functions and Jacobi forms, the previous D-finite functions being (relatively
simple) rational functions of Jacobi theta functions, when rewritten in terms
of the nome of elliptic functions.
