On the asymptotics of a sequence of lacunary binomial-type polynomials

Abstract

We examine the asymptotics of a sequence of lacunary binomial-type polynomials ℘n(z) as n → ∞ that have arisen in the problem of the expected number of independent sets of vertices of finite simple graphs. We extend the recent analysis of Gawronski and Neuschel by employing the method of steepest descents applied to an integral representation. The case of complex z with |z| < 1 is also considered. Numerical results are presented to illustrate the accuracy of the resulting expansions.Mathematics Subject Classification: 30E15, 41A60

$PC$-polynomial of graph

“…In [41], it was proven that ln C(G n,p , 1) ∼ ln 2 n 2 ln(1/p) . In 2014, W. Gawronski and T. Neuschel stated the following result: (see also [163]) Theorem 10.3 [97]. For a fixed p ∈ (0; 1), as n → ∞, we have…”

“…There are some articles devoted to the study of the behavior and properties of C(G n,p , 1). In 2014, W. Gawronski and T. Neuschel stated (see also [163])…”