Infra logarithm and ultra power part functions

Abstract

Considering the serial operations as multiplication and power, we introduced the next step of them in [M.H. Hooshmand, Ultra power and ultra exponential functions, Integral Transforms Spec. Funct. 17 (2006), pp. 549–558], and called it ultra power. In that way, we proved a uniqueness theorem for the (ultra exponential) functional equation f(x)=a f(x−1), x>−1, with the initial condition f(0)=1, and got the new function uxp a for which uxp a (n)=a n (a to the ultra power of n) for every n∈ℕ. For this reason, we call uxp a ultra exponential function. In this paper, we want to introduce another new function Iog a , namely infra logarithm function, that is the dual of uxp a and for a>1 is its inverse. We show that Iog a satisfies the dual of the ultra exponential equation that is the same Abel's exp a -functional equation (f(a x )=f(x)+1), where x runs over ℝ\\ [δ1, δ2] (δ1, δ2 will be introduced). For this reason, sometimes we call it infra logarithm functional equation. There after, we prove that Iog a is the unique solution of the Abel's equation, under some conditions, and show that it together with ultra power part function (another new function) are the essential solutions of it, and then determine the general solution of the Abel's exp a -equation for every 0<a≠1. Also we state another functional equation f(a x )=f((x))+[x]+1, namely co-infra logarithm functional equation, and prove that it is equivalent to the Abel's equation when x runs over . Moreover, we introduce some properties of the infra logarithm function and its relations to the ultra exponential function and their related equations.