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Strict differentiability and \(q\)-expansion of functions on \(\mathbb Z_p\) when \(q\) is a primitive \(p^N\)-th root of unity. (English) Zbl 1185.11075

Let \(K\) be a complete valued field extension of \(\mathbb Q_p\), \(q\in K\), \(|q-1|<1\). A \(q\)-analogue of the Mahler expansion of a continuous function on \(\mathbb Z_p\) with values in \(K\) was constructed by K. Conrad [Adv. Math. 153, No. 2, 185–230 (2000; Zbl 1003.11055)]. The author considers a subclass of strictly differentiable functions [see W. H. Schikhof, Ultrametric calculus. An introduction to \(p\)-adic analysis. Cambridge etc.: Cambridge University Press (1984; Zbl 0553.26006)] and refers to his preprint “On the strictly differentiable functions on \(\mathbb Z_p\)” (2006) for the description of this class in terms of coefficients of the \(q\)-Mahler expansion, provided \(q\) is not a root of unity.
In the paper under review the author gives such a description for the case where \(q\) is a primitive \(p^N\)-th root of unity. A calculation of the Volkenborn integral of a function on \(\mathbb Z_p\) in terms of the same coefficients is also given.

MSC:

11S80 Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.)
46S10 Functional analysis over fields other than \(\mathbb{R}\) or \(\mathbb{C}\) or the quaternions; non-Archimedean functional analysis
26E30 Non-Archimedean analysis
05A30 \(q\)-calculus and related topics
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