Benharrat Belaïdi, Rabab Bouabdelli and Abdelbaki Boutabaa: Ultrametric $q$-difference equations and $q$-Wronskian, p.137-145

Abstract:

Let Κ be an ultrametric complete and algebraically closed field and let $q$ be an element of Κ which is not a root of unity and is such that $\vert q\vert=1$. In this article, we establish some inequalities linking the growth of generalized $q$-wronskians of a finite family of elements of Κ$\KK[[x]]$ to the growth of the ordinary $q$-wronskian of this family of power series.
We then apply these results to study some $q$-difference equations with coefficients in Κ$\KK[x]$. Specifically, we show that the solutions of such equations are rational functions.

Key Words: Ultrametric, $p$-adic, $q$-difference equation, $q$-wronskian.

2000 Mathematics Subject Classification: Primary: 12J25
Secondary: 46S10.

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