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D-finite power series. (English) Zbl 0695.12018
A power series $$f(x)=\sum a_ ix^ i$$ is called D-finite if all the derivatives of f span a finite-dimensional vector space over $${\mathbb{C}}(x)$$. A sequence $$(a_ i)$$ is called P-recursive if it satisfies a recursion of the form $$p_ d(i)a_ i+p_{d-1}(i)a_{i-1}+...+p_ 0(i)a_{i-d}=0$$ where the $$p_ j(i)$$ are polynomials. The connection between the two concepts is that $$\sum a_ ix^ i$$ is D-finite if and only if $$(a_ i)$$ is P-recursive.
The concepts of D-finiteness and P-recursiveness are generalized to power series in several variables. A number of results about D-finite power series and P-recursive sequences is given.
A Hartogs’-type theorem for D-finite analytic functions f(x,y) is proved: if the restriction of f to each line segment is D-finite as a function of one variable, then f is D-finite as a function of two variables. It is proved that if the infinite matrix $$(a_{ij})_{i,j\in {\mathbb{N}}}$$ has the properties that (i) each row contains only finitely many nonzero entries and (ii) for every P-recursive sequence $$(b_ j)$$ the matrix product $$(a_{ij})(b_ j)=(\sum_{j}a_{ij}b_ j)$$ is P-recursive, then $$(a_{ij})$$ is P-recursive.
Reviewer: E.V.Pankrat’ev

##### MSC:
 12H20 Abstract differential equations 13F25 Formal power series rings 32A05 Power series, series of functions of several complex variables
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