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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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