Asymptotic behavior of high-order differences of the plane partition function (Q1318806)

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.