The \(p\)-adic \(z\)-transform (Q1898748)

Let \(a+ p^n \mathbb{Z}_p\) be a ball in \(\mathbb{Z}_p\). For a given \(z\in \mathbb{C}_p\) let \({\mathfrak m}_z (a+p^n \mathbb{Z}_p)= z^a/ (1- z^{p^n})\), where \(z\in \mathbb{C}_p\), \(|z-1 |_p\geq 1\). The \(p\)-adic distribution \({\mathfrak m}_z\) defines a kind of \(z\)-transform by \(F(z)= \int_{\mathbb{Z}_p} f(x) {\mathfrak m}_z (x)\). The author presents a systematic study of this transform. Moreover he gives some applications, e.g. to Mahler's theorem on representations of continued functions \(f: \mathbb{Z}_p\to \mathbb{Q}_p\) in the form \(f(x)= \sum^\infty_{n=0} a_n \binom x n\), van der Put expansions and to the expansion of a \(p\)-adic function in a series of Sheffer polynomials.