Class of non-associative functions of a single positive real variable

Abstract

A function F/sub n,..cap alpha../(x) = XI/sub j=1//sup n/(g/sub j/(x))/sub ..cap alpha../ has been defined which is a generalization of the function f(x) = x/sup x/sup x...// with an infinite number of x's. The value of F/sub n,..cap alpha../(x) depends on the bracketing order, which is denoted by ..cap alpha.. (non-associativity), and on the number, n, of functions g/sub j/(x) included in the powering operation. The purpose of this paper is to study the limit of F/sub n,..cap alpha../(x) as n ..-->.. infinity for a number of choices of g/sub j/(x) which have been proposed in the earlier paper, in particular, g/sub j//sup (A)/(x) = x/j, g/sub j//sup (B)/(x) = x/j/sup 2/, g/sub j//sup (D)/(x) = x/2/sup j/, and g/sub j//sup (F)/(x) = x/j/sup /sup 1///sub 2//. The bracketing order ..cap alpha.. considered in the main part of this paper is the bracketing (order of parentheses) denoted by a in the previous work. Several interesting properties of the functions x/sup y/ and E(x,y) identical with x/sup y/ + y/sup x/ are discussed in an appendix. 9 figures, 5 tables.