Partition polynomials: asymptotics and zeros

Abstract

Let F n (x) be the partition polynomial ∑ n k=1 p k (n)x k where p k (n) is the number of partitions of n with k parts. We emphasize the computational experiments using degrees up to 70, 000 to discover the asymptotics of these polynomials. Surprisingly, the asymptotics of F n (x) have two scales of orders n and √ n and in three different regimes inside the unit disk. Consequently, the zeros converge to network of curves inside the unit disk given in terms of the dilogarithm.

Asymptotics for the twisted eta-product and applications to sign changes in partitions

“…, so it suffices to find asymptotic formulas for ζ in the upper-half plane. The following theorem of Boyer and Goh [5] regarding the dilogarithm distinguishes several cases in our main theorem. Following [5], we define…”

“…The following theorem of Boyer and Goh [5] regarding the dilogarithm distinguishes several cases in our main theorem. Following [5], we define…”

“…Figure 2 shows that this prediction is surprisingly accurate even for small n. The proof makes use of (1.4) and a detailed study of the coefficients n≥0 Q n (ζ)q n := (ζq; q) −1 ∞ when ζ is a root of unity. Boyer and Goh [5] studied the growth of these coefficients for |ζ| < 1, and showed that in this case, the open unit disk can be divided into three regions such that the growth is uniform on compact subsets of each region. Asymptotics for the coefficients of (ζq; q) −1 ∞ for |ζ| < 1 were also studied in 2017 by Parry [16] in a more general setting, and the case that ζ is any positive real number was studied Wright [21].…”

“…have the same growth in their amplitudes but have different phases of sign changes. Since P S 1 ,S 2 (x) has zeros {1, ±ζ 12 , ±ζ 5 12 }, the dominating term in the asymptotic expansion is induced by the root of unity ζ 6 . In fact, we obtain ∆ r (n)…”

Solving polynomial eigenvalue problems by means of the Ehrlich–Aberth method

“…Eigenvalue R. e., polyeig R. e., eig R. e., UEAI R. e., SEAI - 3.2e-7 2.2e-9 6.1e-12 1.5e-11 7 3.1e-6 3.3e-9 1.5 e-11 1.6e-11 8 4.6e-6 2.7e-9 1.5e-11 5.5e-12 9 2.9e-6 1.1e-9 1.7e-12 2.5e-12 10 4.2e-6 1.7e-10 3.0e-12 6.9e-13 Average 3.8e-5 9.6e-10 5.5e-12 4.1e-13…”