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{\bf Philippe Flajolet, Stefan Gerhold and Bruno Salvy}
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{\bf On the Non-Holonomic Character of Logarithms, Powers, and the $n$th
Prime Function}
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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 [{\it El. J. Comb.} {\bf 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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