Iterates of the Exponential Function and an Ingenious Formal Technique

Abstract

The first seven paragraphs of Chapter 4 are concerned with iterated exponential functions and constitute a sequel to a large portion of Chapter 3 wherein the Bell numbers, single-variable Bell polynomials, and related topics are studied. Recall that the Bell numbers B(n), 0 ≤ n ≤ ∞, may be defined by They were first thoroughly studied in print by Bell [1], [2] approximately 25–30 years after Ramanujan had derived several of their properties in the notebooks. Further iterations of the exponential function appear to have been scarcely studied in the literature. The most extensive study was undertaken by Bell [2] in 1938. Becker and Riordan [1] and Carlitz [1] have established arithmetical properties for these generalizations of Bell numbers. Also, Ginsburg [1] has briefly considered such iterates. For a combinatorial interpretation of numbers generated by iterated exponential functions, see Stanley’s article [1, Theorem 6.1].