On Moessner’s Theorem

Moessner’s theoremdescribes a procedure for generating a sequence ofninteger sequences that lead unexpectedly to the sequence ofnth powers 1 n , 2 n , 3 n , …Paasche’s theoremis 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 sequencea· 1 n−1, (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.