Three generalizations of Mahler's expansion for continuous functions on ℤp

A q-Analogue of Mahler Expansions, I

“…In [6], a q-analogue of Mahler expansions will be described for q # K, |q| =1, that will reduce to van Hamme's expansion when q # Z _ p and q is not a root of unity. In [24,Theorem 3], van Hamme gives a remainder formula for the Mahler expansion. For a complete extension field KÂQ p and a continuous function f : Z p Ä K with Mahler coefficients c n ,…”

The p-adic Z-transform

“…> 1 and use (5) for the first integral and (3) for the second integral. The formula then reduces to the obvious identity.…”

“…This was obtained in [5] by a different method. In this section we will use the p-adic z-transform to generalize the main theorem of [6].…”