On Moessner's Theorem

Abstract

Moessner's theorem describes a procedure for generating a sequence of n integer sequences that lead unexpectedly to the sequence of nth powers 1^n, 2^n, 3^n, ... Paasche's theorem is a generalization of Moessner's; by varying the parameters of the procedure, one can obtain the sequence of factorials 1!, 2!, 3!, ... or the sequence of superfactorials 1!!, 2!!, 3!!, ... Long's theorem generalizes Moessner's in another direction, providing a procedure to generate the sequence a, (a+d)2^{n-1}, (a+2d)3^{n-1}, ... Proofs of these results in the literature are typically based on combinatorics of binomial coefficients or calculational scans. In this note we give a short and revealing algebraic proof of a general theorem that contains Moessner's, Paasche's, and Long's as special cases. We also prove a generalization that gives new Moessner-type theorems.