Propriétés algébriques de suites différentiellement finies. (Algebraic properties of differentially finite sequences). (French) Zbl 0785.11007

Benzaghou, Benali; Bézivin, Jean-Paul

Bull. Soc. Math. Fr. 120, No. 3, 327-346 (1992).

A sequence \((u_ n)\) over a field \(K\) is said to be differentially finite if the power series \(\sum u_ n x^ n\) satisfies a linear differential equation over \(K(x)\). If \(\sum u_ n x^ n\) is rational then \((u_ n)\) is said to be a recurrence sequence. The authors study the situations in which both \((u_ n)\) and \((1/u_ n)\) are \(D\)-finite or in which the \(u_ n\) are zeros of a polynomial equation with recurrence sequence coefficients.
It is well known that both \((u_ n)\) and \((1/u_ n)\) are recurrence sequences only if \((u_ n)\) is composed of finitely many subsequences consisting just of geometric progressions. In that spirit the authors conjecture that both \((u_ n)\) and \((1/u_ n)\) are \(D\)-finite only if \((u_ n)\) is composed of finitely many subsequences consisting just of hypergeometric progressions and provide some evidence supporting that suggestion. In particular if \((u_ n)\) satisfies a polynomial equation with recurrence sequence coefficients and \((1/u_ n)\) is \(D\)-finite then \((u_ n)\) is composed of finitely many subsequences consisting just of hypergeometric progressions.

Reviewer: A.J.van der Poorten (North Ryde)

MSC:

11B37	Recurrences
12H05	Differential algebra
12H10	Difference algebra

Keywords:

differentially finite sequences; recurrence sequence

References:

[1]	BENZAGHOU (B.) . - Algèbres de Hadamard , Bull. Soc. Math. France, t. 98, 1970 , p. 209-252. Numdam | MR 44 #1658 | Zbl 0206.33203 · Zbl 0206.33203
[2]	BEZIVIN (J.-P.) . - Sur un théorème de G. Polya , J. Reine Angew. Math., t. 364, 1986 , p. 60-68. MR 87f:11053 | Zbl 0569.10004 · Zbl 0569.10004
[3]	BEZIVIN (J.-P.) . - Une généralisation du théorème de Skolem-Mahler-Lech , Quart. J. Math. Oxford, t. 40, 1989 , p. 133-138. MR 90g:11021 | Zbl 0678.10040 · Zbl 0678.10040
[4]	BEZIVIN (J.-P.) . - Quotient de fonctions entières et quotients de Hadamard de séries formelles , Ann. Inst. Fourier, t. 39, 1989 , p. 737-752. Numdam | MR 90k:30002 | Zbl 0701.30004 · Zbl 0701.30004
[5]	CIERLENCO (I.) , MIGNOTTE (M.) et PIRAS (F.) . - Suites récurrentes linéaires : propriétés algébriques et arithmétiques , Enseign. Maths, t. 33, 1987 , p. 67-108. MR 88h:11010 | Zbl 0626.10008 · Zbl 0626.10008
[6]	JUNGEN (R.) . - Sur les séries de Taylor n’ayant que des singularités algébrico-logarithmiques sur leur cercle de convergence , Comment. Math. Helv., t. 3, 1931 , p. 266-306. Zbl 0003.11901 | JFM 57.0373.03 · Zbl 0003.11901
[7]	POLYA (G.) . - Arithmetische Eigenschaften der Reihenenwicklungen rationaler Funktionen , J. Reine Angew. Math., t. 151, 1921 , p. 1-31. JFM 47.0276.02 · JFM 47.0276.02
[8]	REUTENAUER (C.) . - Sur les éléments inversibles de l’algèbre de Hadamard des séries rationnelles , Bull. Soc. Math. France, t. 110, 1982 , p. 225-232. Numdam | MR 84f:16001 | Zbl 0517.13010 · Zbl 0517.13010
[9]	RITT (J.-F.) . - Algebraic combination of polynomials , Trans. Amer. Math. Soc., t. 31, 1929 , p. 654-689. MR 1501505 | JFM 55.0211.03 · JFM 55.0211.03
[10]	RUMELY (R.) and VAN DER POORTEN (A.) . - A note on the k-th root of a rational function , J. Austr. Math. Soc., t. 43, 1987 , p. 314-327. MR 88j:11009 | Zbl 0635.10008 · Zbl 0635.10008
[11]	STANLEY (R.P.) . - Differentially finite power series , European J. Combin., t. 1, 1980 , p. 175-188. MR 81m:05012 | Zbl 0445.05012 · Zbl 0445.05012