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The diagonal of a D-finite power series is D-finite. (English) Zbl 0657.13024
Let $$K[[x]]$$ be the ring of formal power series in variables $$x_ 1,...,x_ n$$ over a field $$K$$ of characteristic zero. A series $$f\in K[[ x]]$$ is called D-finite (or differentiably finite) if the set of all its derivatives lie in a finite-dimensional vector space over $$K(x)$$, the field of rational functions in $$x_ 1,...,x_ n$$. If $$f=\sum a_{i_ 1...i_ n}x_ 1^{i_ 1}...x_ n^{i_ n}$$ the primitive diagonal $$I_{12}(f)$$ is defined by the formula $I_{12}(f)=\sum a_{i_ 1i_ 3...i_ n}x_ 1^{i_ 1}x_ 3^{i_ 3}...x_ n^{i_ n}.$ The other primitive diagonals $$I_{ij}$$ (for $$i<j$$) are defined similarly. Any composition of the $$I_{ij}$$ is called diagonal.
It is proved that any diagonal of a D-finite power series is again D-finite.

##### MSC:
 13F25 Formal power series rings 13J05 Power series rings 05A15 Exact enumeration problems, generating functions
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