D-finite power series.

*(English)*Zbl 0695.12018A 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.

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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