A generalisation of the functions г( n ) and x n .

Abstract

It is interesting to develope from simple principles a generalisation of the functions x n and г( n ). Consider an infinite sequence (1, p , p 2 , p 3 , . . . p n , . . .), the write [1] = 1, [2] = 1+ p , . . . . . . [ n ] = 1 + p + p 2 + . . . + p n -1 ( n positive and integral), [- n ] = - p -1 - p -2 - . . . - p - n ( n integral).