  The article is based on the differential properties of the inner generators
(singularities) that occur while handling SP series (sum product series), power
series whose Taylor coefficients can be written as sum-product combinations. It
is an elaboration with numerical details and some explicit calculations of the
chapter 6 of the joint article with Ecalle on these series. For all polynomial
inputs $f$ and all monomial inputs $F$, the inner generators corresponding to
them verify ordinary differential equations of linear homogeneous type with
polynomial coefficients. Numerical results strongly suggest the non-existence
of ODEs for functions $F$ that are rational but not monomial, i.e. not of the
form ${(1-ax)}^p$, for $p \in \mathbb{Z}$.
