A \(q\)-analogue of Mahler expansions. I
The author studies a \(q\)-analogue of Mahler expansions for continuous functions \(f:\;\mathbb Z_p\to \mathbb Q_p\). The binomial coefficient polynomials \(\binom xn\) are replaced with their \(q\)-analogue \(\binom xn_q\), \(q\in \mathbb Z_p\), \(|q-1|_p<1\). Four different proofs of the expansion theorem are given. A formula for the coefficients in the expansion of \(f\) is obtained, as well as characterizations of differentiability and analyticity in terms of the above coefficients, and a formula for the expansion into a finite part of the \(\binom xn_q\)-series and a remainder term. An interpretation of the \(p\)-adic interpolation of Koblitz's \(p\)-adic \(q\)-Gamma function in terms of its \(q\)-Mahler expansion is proposed. Earlier \(\binom xn_q\)-expansions were obtained for functions defined on certain closed subsets of \(\mathbb Z_p\); see \textit{Y. Amice} [Bull. Soc. Math. Fr. 92, 117--180 (1964; Zbl 0158.30203)], \textit{L. van Hamme} [Proc. Conf. on \(p\)-adic Analysis, Nijmegen, 119--125 (1978)], and \textit{A. Verdoodt} [Indag. Math., New Ser. 4, 375--384 (1993; Zbl 0788.11056)].
scientific article; zbMATH DE number 1507786
scientific article