On the Non-Holonomic Character of Logarithms, Powers, and the nth Prime Function

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 [El. J. Comb. 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. Es ist eine Tatsache, daß die genauere Kenntnis des Verhaltens einer analytischen Funktion in der Nähe ihrer singulären Stellen eine Quelle von arithmetischen Sätzen ist.1 — Erich Hecke [25, Kap. VIII]