On Moessner’s Theorem
Dexter C. Kozen, Alexandra Silva · American Mathematical Monthly · 2013
Moessner's theorem describes a procedure for generating a sequence of n integer sequences that lead unexpectedly to the sequence of nth powers 1n, 2n, 3n, … Paasche's theorem is a generalization of Moessner's; by varying the parameters of the procedure, we can obtain the sequence of factorials 1!, 2!, 3!, … or the sequence of superfactorials 1!, 2! 1!, 3! 2! 1!, … Long's theorem generalizes Moessner's in another direction, providing a procedure to generate the sequence a · 1n−1, (a + d) · 2n−1, (a + 2d) · 3n−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.