Applied hypernumbers: computational concepts

摘要

The key importance of hypernumbers in enlarging and fruitfully generalizing (as distinct from abstraction of a sterile sort) algebra, function theory and computation is discussed, with specific examples and theorems. The rich serendipity of hypernumber research is shown in the author's recent findings; for example, those generalizing the Bernoulli numbers for any real, complex, or countercomplex index s, as Bs= -s!2cos(πs/2) ζ(s)/(2π)s, where ζ is Riemann's Zeta function; whence, e.g., B0=1, B2=1/6, B1/2=hf;ζ(hf;), and B3/B5=-(80π2)-1ζ(3)/ζ(5), results like the last two being unknown and unobtainable before. As in APL computer language, the symbol “!” is used to denote Gauss' π function: the factorial of unrestricted argument.