partition.fn.diff

DD-‡D~-pDD-Differences of the partition functionA. M. OdlyzkoAT&T Bell LaboratoriesMurray Hill, New Jersey 07974ABSTRACTLet p(n) denote the number of unrestricted partitions of n, and letk k 1p(n) = p(n) p(n 1 ), p(n) = D ( p ) (n). This note answers severalkquestions about the behavior of the k-difference p(n) by proving that if k is largekenough, there is an integer n (k) such that p(n) alternates in sign for n < n (k)0 06 2 2_ __and is nonnegative for n n (k). It is also shown that n (k) k ( log k) as0 0 2k fi ¥ .D-DD-DDp--D-D‡D‡DD‡DDDDifferences of the partition functionA. M. OdlyzkoAT&T Bell LaboratoriesMurray Hill, New Jersey 07974. .Dedicated to Paul Erdos on the occasionof his 75th Birthday.1. IntroductionIf f (n) is any function on the nonnegative integers, define its first difference f bykf (n) = f (n) f (n 1 ) for n 1, f ( 0 ) = f ( 0 ). The k-th difference f of f isk k 1then defined recursively by f = D ( f). A few years ago, I. J. Good [5a] askedkabout the behavior of p(n), where p(n) denotes the number of unrestricted partitionskof n. He initially conjectured [5a] that if k > 3, then the sequence p(n),n = 0 , 1 , . . . , alternates in sign. However, computations by R. Razen andindependently by I. J. Good and his associates [5b] found counterexamples to thiskconjecture, and led to a new conjecture, namely that for each fixed k, p(n) > 0 for nsufficiently large. I. J. Good [5b] even made the stronger ...