Difference-differential fields of continuous functions

TL;DR

This paper explores the algebraic structures of continuous functions on [0,∞), introducing new difference and q-difference operators that extend Mikusinski's operational calculus to include q-shift and Mahler-type difference fields.

Contribution

It defines novel transforming operators related to q-shift and Mahler difference operators, expanding the algebraic framework of Mikusinski's operational calculus.

Findings

Established a q-difference field structure on Q(C)

Developed a Mahler-type difference field structure

Revisited and extended Mikusinski's operational calculus

Abstract

The set C of complex-valued continuous functions on [0,\infty) is a ring by the addition and the convolution. It has the quotient field Q(C), by which J. Mikusinski developed his operational calculus. In this paper, we revisit a derivation and a transforming operator for Q(C) written in his textbook, and define another transforming operator related to the q-shift operator, which gives structures of a q-difference field and a difference field of Mahler type to Q(C). Appropriate derivatives are also considered.