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On the non-holonomic character of logarithms, powers, and the n-th prime function

Philippe Flajolet·Stefan Gerhold·Bruno Salvy
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Abstract

We establish that the sequences formed by logarithms and by "fractional" powers of integers, as well as the sequence of prime numbers, are non-holonomic, thereby answering three open problems of Gerhold [Electronic Journal of Combinatorics 11 (2004), R87]. Our proofs depend on basic complex analysis, namely a conjunction of the Structure Theorem for singularities of solutions to linear differential equations and of an Abelian theorem. A brief discussion is offered regarding the scope of singularity-based methods and several naturally occurring sequences are proved to be non-holonomic.

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Philippe Flajolet, Stefan Gerhold, Bruno Salvy. 2005-01-22. On the non-holonomic character of logarithms, powers, and the n-th prime function. https://arxiv.org/abs/math/0501379

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