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

##### MSC:
 11B37 Recurrences 12H05 Differential algebra 12H10 Difference algebra
##### Keywords:
differentially finite sequences; recurrence sequence
Full Text:
##### References:
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