Jackson's formula with remainder in \(p\)-adic analysis
Let \(f: \mathbb{Z}_ p\to \mathbb{C}_ p\) (where \(\mathbb{Z}_ p\) is the set of the \(p\)-adic integers and where \(\mathbb{C}_ p\) is the completion of the algebraic closure of the \(p\)-adic field). It was proved by van Hamme that in the Mahler expansion of \(f\) \[ f(x)= \sum_{i=0}^ n (\Delta^ i f)(0) {x \choose i} +R_ n(x) \] (where \(\Delta f(v)= f(v+1)- f(v)\)) the remainder \(R_ n\) is some convolution of \(x\mapsto {x\choose n}\) and \(\Delta^{n+1}f\). In this paper the result is generalized to continuous \(f: V_ q\to \mathbb{C}_ p\) (here \(V_ q:= \overline {\{aq^ n:\;n\in\mathbb{N}\}}\) where \(a,q\in\mathbb{Z}_ p\), \(| a|= | q|=1\), \(q\) not a root of unity) by using \(q\)-calculus. One obtains the expression \[ f(x)= \sum_{i=0}^ n\;{{D_ q^ i f(a)} \over {[i]!}} (x-a)^{(i)}+ \widetilde {R}_ n(x) \] where \(\widetilde{R}_ n\) is some convolution involving \({{(x-a)^{(n)}} \over {[n]!}}\) and \(D_ q^{n+1}f\). It is also proved that \(\lim_{n\to\infty} \widetilde {R}_ n(x)=0\) yielding the well-known Jackson formula \[ f(x)=\sum_{i=0}^ \infty {{D_ q^ i f(a)} \over {[i]!}} (x-a)^{(i)}. \] {}.
\(p\)-adic Jackson formula with remainder
scientific article; zbMATH DE number 510033