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

Flajolet, Philippe; Gerhold, Stefan; Salvy, Bruno

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

Authors

Flajolet, Philippe

Gerhold, Stefan

Salvy, Bruno

Description

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.
13 pages