TELECOM -Italia, Via delle Azzorre 352-D2, 00121 Roma, Italy
(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.