Geodesic
On some non-holonomic sequences
The electronic journal of combinatorics, Tome 11 (2004) no. 1

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Zbl EuDML
A sequence of complex numbers is holonomic if it satisfies a linear recurrence with polynomial coefficients. A power series is holonomic if it satisfies a linear differential equation with polynomial coefficients, which is equivalent to its coefficient sequence being holonomic. It is well known that all algebraic power series are holonomic. We show that the analogous statement for sequences is false by proving that the sequence $\{\sqrt{n}\}_n$ is not holonomic. In addition, we show that $\{n^n\}_n$, the Lambert $W$ function and $\{\log{n}\}_n$ are not holonomic, where in the case of $\{\log{n}\}_n$ we have to rely on an open conjecture from transcendental number theory.
DOI : 10.37236/1840
Classification : 11B37, 11J81
Mots-clés : holonomic sequence, holonomic power series, algebraic power series 
Stefan Gerhold. On some non-holonomic sequences. The electronic journal of combinatorics, Tome 11 (2004) no. 1. doi: 10.37236/1840
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     author = {Stefan Gerhold},
     title = {On some non-holonomic sequences},
     journal = {The electronic journal of combinatorics},
     year = {2004},
     volume = {11},
     number = {1},
     doi = {10.37236/1840},
     zbl = {1063.11007},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/1840/}
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