SOME MOBIUS-TYPE FUNCTIONS AND INVERSIONS CONSTRUCTED VIA DIFFERENCE OPERATORS

It is shown that some difference operators and their inverses, defined on the hyper-real field *$\mathbb{R}$ can be used to generate a pair of reciprocal relations that implies both the M\''{o}bius inversion formulae and the fundamental theorem of calculus as special consequences. As suggested by the form for the M\''{o}bius function of integral order, some explicitly con_structive extensions of Mi:ibius-type functions are presented; and accordingly, certain general M\''{o}bius-type inversion pairs are obtained in a natrural way.