×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Björk, J.E, Rings of differential operators, (1979), North-Holland Amsterdam/Oxford/New York
[2] Denef, J; Lipshitz, L, Algebraic power series and diagonals, J. number theory, 26, 46-67, (1987) · Zbl 0609.12020
[3] \scI. Gessel, Symmetric functions and p-recursiveness, preprint.
[4] Lipshitz, L, The diagonal of a D-finite power series is D-finite, J. algebra, 113, 373-378, (1988) · Zbl 0657.13024
[5] Mahler, K, Lectures on transcendental numbers, Springer lecture notes, Vol. 546, (1976), Berlin/Heidelberg/New York · Zbl 0213.32703
[6] Ostrowski, A, Über dirichletsche reihen und algebraische differentialgleichungen, Math. Z., 8, 241-298, (1920) · JFM 47.0292.01
[7] Palais, R, Some analogous of Hartogs’ theorem in an algebraic setting, Amer. J. math., 100, 387-405, (1978) · Zbl 0449.14002
[8] Stanley, R.P, Differentiably finite power series, European J. combin., 175-188, (1980) · Zbl 0445.05012
[9] Zeilberger, D, Sister Celine’s technique and its generalizations, J. math. anal. appl., 85, 114-145, (1982) · Zbl 0485.05003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.