Kim, Taekyun; Kim, Seung Dong; Park, Dal-Won On uniform differentiability and \(q\)-Mahler expansions. (English) Zbl 1003.11056 Adv. Stud. Contemp. Math., Pusan 4, No. 1, 35-41 (2001). 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. Reviewer: Anatoly N.Kochubei (Kyïv) Cited in 8 Documents 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 Keywords:\(q\)-Mahler expansion; \(q\)-binomial coefficients; strictly differentiable function Citations:Zbl 1003.11055 PDFBibTeX XMLCite \textit{T. Kim} et al., Adv. Stud. Contemp. Math., Pusan 4, No. 1, 35--41 (2001; Zbl 1003.11056)