Partition polynomials and their zeros (Q2774481)

Define \(\omega(k):=(3k^2-k)/2\). The main object of study in this paper are the \textit{partition polynomials} NEWLINE\[NEWLINE\mathcal P_n(x):=x^{\omega(n)}+\sum_{k=1}^{n-1}(-1)^k \left[x^{\omega(n)-\omega(k)}+x^{\omega(n)-\omega(-k)}\right]+(-1)^nNEWLINE\]NEWLINE (the name arises from the fact that these are the characteristic polynomials which arise from Euler's recurrence for the ordinary partition function \(p(n)\)). A number of results on the distribution of the zeros of the polynomials \(\mathcal P_n(x)\) are given. For example, it is shown that the zeros of \(\mathcal P_n(x)\) tend to the unit circle as \(n\to\infty\); in particular it is shown that there exist positive constants \(c_1\) and \(c_2\) such that for all \(n\geq 1\) we have NEWLINE\[NEWLINE1-c_1/n<|x|<1+c_2/nNEWLINE\]NEWLINE for every zero \(x\) of \(\mathcal P_n(x)\). Estimates for the largest real zero are given which show that the upper bound is best possible. Several very precise conjectures concerning the distribution of these zeros are also given.