Asymptotics and zero distribution of geometric polynomials

Abstract

We obtain some results on the asymptotic behavior and zero distribution of the so-called geometric polynomials. The asymptotics is given both on compact subsets of $\C\setminus [-1,0]$ and on compact subsets of the interval $(-1,0)$. The zeros of these polynomials are simple and lie in $(-1,0]$; moreover, the zeros of consecutive polynomials interlace. Its zero distribution is a measure whose density is similar to Cauchy weight. Some orthogonality properties of these polynomials are also proved.

Figures (2)

• Figure 1: Graphics of $P_n(z)$ for $n=0,1,\ldots,6$ in the interval $[-1.01,0.1]$.
• Figure 2: Graphics of $\frac{(z+1)P_5(z)(\ell(z)+\pi^2)^3}{3!}$ (left) and its approximation, $\frac{(\ell(z)+\pi i)^6+(\ell(z)-\pi i)^6}{(\ell(z)+\pi^2)^3}$ (right).

Theorems & Definitions (21)

• Theorem 1
• Theorem 2
• Lemma 1
• proof
• Remark 1
• Lemma 2
• proof
• Lemma 3
• proof
• proof : Proof of Theorem \ref{['teoAsymp']}