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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
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