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

Power series with Taylor coefficients of sum-product type and algebraic  differential equations /SP-series and the interplay between their resurgence  and differential properties