Geodesic
On the non-holonomic character of logarithms, powers, and the \(n\)th prime function
The electronic journal of combinatorics, The Stanley Festschrift volume, Tome 11 (2004) no. 2
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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 [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.
DOI : 10.37236/1894
Classification : 05A15, 11B83, 33E30
Mots-clés : holonomic sequence, \(P\)-recursive sequence, generating function, singularities 
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     author = {Philippe Flajolet and Stefan Gerhold and Bruno Salvy},
     title = {On the non-holonomic character of logarithms, powers, and the \(n\)th prime function},
     journal = {The electronic journal of combinatorics},
     year = {2004},
     volume = {11},
     number = {2},
     doi = {10.37236/1894},
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     url = {http://geodesic.mathdoc.fr/articles/10.37236/1894/}
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Philippe Flajolet; Stefan Gerhold; Bruno Salvy. On the non-holonomic character of logarithms, powers, and the \(n\)th prime function. The electronic journal of combinatorics, The Stanley Festschrift volume, Tome 11 (2004) no. 2. doi: 10.37236/1894

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