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On uniform differentiability and \(q\)-Mahler expansions. (English) Zbl 1003.11056

It was proved by K. Conrad [Adv. Math. 153, 185-230 (2000; Zbl 1003.11055)] that the \(q\)-binomial coefficient functions \(\binom xn_q\), where \(q\in \mathbb Z_p\), \(|q-1|_p<1\), form an orthonormal basis in the space of continuous functions on \(\mathbb Z_p\) with values from an extension \(K\) of \(\mathbb Q_p\). A characterization of the expansion coefficients for differentiable functions was also given.
The authors introduce the notion of a strictly differentiable function \(f:\;\mathbb Z_p\to K\). That means that \((x-y)^{-1}[f(x)-f(y)]\) has a limit as \((x,y)\to (a,a)\), for every \(a\in \mathbb Z_p\). Let \(C^{(m)}(\mathbb Z_p,K)\) be the space of \(m\) times strictly differentiable functions. Their characterization is given in terms of coefficients of \(\binom xn_q\)-expansions. It is shown that the \(q\)-binomial coefficient functions form an orthonormal basis in \(C^{(m)}(\mathbb Z_p,K)\) with respect to the natural norm.

MSC:

11S80 Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.)
11B65 Binomial coefficients; factorials; \(q\)-identities
12J25 Non-Archimedean valued fields

Citations:

Zbl 1003.11055
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