Open AccessSOME MOBIUS-TYPE FUNCTIONS AND INVERSIONS CONSTRUCTED VIA DIFFERENCE OPERATORS
Author(s) -
L. C. Hsu,
Jun Wang
Publication year - 1998
Publication title -
tamkang journal of mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.324
H-Index - 18
eISSN - 0049-2930
pISSN - 2073-9826
DOI - 10.5556/j.tkjm.29.1998.4278
Subject(s) - mathematics , reciprocal , type (biology) , inversion (geology) , pure mathematics , order (exchange) , field (mathematics) , algebra over a field , calculus (dental) , ecology , biology , medicine , paleontology , philosophy , linguistics , dentistry , finance , structural basin , economics
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.