2005/04/28 by Philippe Flajolet, Stefan Gerhold, Bruno Salvy · 1 citation
Mathematics · #Abelian group #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Algebra over a field #Character (mathematics) #Combinatorics #Discrete mathematics #Generating function #Holonomic #Logarithm #Mathematical analysis #Mathematical proof #Mathematics #Mathematics and Applications #Prime (order theory) #Pure mathematics
paper · pdf · doi:10.37236/1894
openalex publication_date 2005/04/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/26
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.