Andrea Ossicini
TELECOM -Italia, Via delle Azzorre 352-D2, 00121 Roma, Italy
(e-mail: a.ossicini@finsiel.it)
(Received March 12, 2005)
Abstract. The purpose
of the work is to furnish a study of a discrete and special
function, discovered by the author and named with the Arabian letter
(Shin), produced by a family of functions. A
fundamental theorem is enunciated: it connects the rational values
of this family with a natural number; to this aim two rational-value
functions will be created, with the characteristic, in the field of
real positive numbers, to be piecewise continuous. We state not only
the existence of a separation element, but we prove that this
element is just formed by one only integer constant function, the
value of which is equal to 2. We point out a hypothetical, subtle
connection among the special function , the Eulerian function
Gamma and the second-order Eulerian numbers. It is finally proved
that is completely monotonic: this characteristic is peculiar
for the functions that have considerable applications in different
fields of pure and applied Mathematics.