Asymptotic behavior of high-order differences of the plane partition function
type: Item
instance of: scholarly article
title: Asymptotic behavior of high-order differences of the plane partition function
zbMATH Open document ID: 0795.05006
DOI: 10.1016/0012-365X(94)90263-1
author: Charles Knessl
published in: Discrete Mathematics
publication date: 1994-04-04T00:00:00Z
review text: Let \(\pi (n)\) be the number of plane partition of \(n\) and let \(\Delta^ k \pi (n)\) be the \(k\)th backward difference of \(\pi (n)\). The author studies the asymptotic behavior of \(\Delta^ k \pi (n)\) when \(n\) and \(k\) are simultaneously large to obtain its asymptotic formula which is valid for \(k=O(n^{2/3})\). It is shown that, asymptotically, \(\Delta^ k \pi (n)\) is positive on a portion of the \((n,k)\) plane and alternates in sign as \(n\) increases. An asymptotic formula is derived for the transition curve that separates the positive and oscillatory regions. Upto leading orders this curve is given by \[ n \sim {2 \over 3} (3C)^{-1/2} (k \log k)^{3/2}, C =\sum^ \infty_{\ell=1} \ell^{-3} = \zeta (3). \] These results are derived by obtaining a recurrence relation for \(\Delta^ k \pi (n)\) and then analyzing it by asymptotic methods such as WKB method and the method of matched asymptotic expansions. Numerical comparisons indicate that \(n_ 0^{\text{WKB}} (k)\) is a very accurate approximation while the leading-order result is very poor.
Mathematics Subject Classification ID: 05A17
zbMATH DE Number: 540935
zbMATH Keywords: high-order differences
zbMATH Keywords: plane partition
zbMATH Keywords: asymptotic behavior
zbMATH Keywords: asymptotic formula
zbMATH Keywords: WKB method
MaRDI profile type: Publication
cites work: On the differences of the partition function
cites work: Q3139640
cites work: Exact asymptotic formulas for the coefficients of nonmodular functions
cites work: Q3669422
cites work: Asymptotic behavior of high‐order differences of the partition function
cites work: The Evaluation of the Constant in the Formula for the Number of Partitions of n
cites work: Differences of the partition function
cites work: ASYMPTOTIC PARTITION FORMULAE
description: scientific article; zbMATH DE number 540935