Asymptotics of a sequence of Witt vectors (Q1196889): Difference between revisions

The paper gives the asymptotics and order estimates of the numbers \(d_ n\) appearing in \[ \prod_{n\geq1}{1\over {1+d_ nt^ n/n!}}=(1-t)e^ t. \] The numbers \(d_ n\) are positive integers. It is shown that, for \(n=2,3,\ldots\), \(d_ n\leq(n-1)!\) when \(n\) is odd, \(d_ n=(n-1)!\), when \(n\) is prime, \(d_ n\geq(n-1)!\), when \(n\) is even, and that \(1-1/n\leq d_ n/(n-1)!\leq1+\alpha_ n/\sqrt{(n)}\), where \(\alpha_ 8=\alpha_{16}=2\), and otherwise \(\alpha_ n=1\). The numbers \(d_ n\) give the dimensions of certain group representations, in which theory also Witt vectors occur.