A family of polynomials with concyclic zeros. II

Abstract:

Let ${\lambda _1}, \ldots ,{\lambda _J}$ be nonzero real numbers. Expand \[ E(z) = \prod {( - 1 + \exp {\lambda _j}z)} ,\] rewrite products of exponentials as single exponentials, and replace every $\exp (az)$ by its approximation ${(1 + a{n^{ - 1}}z)^n}$, where $n \geqslant J$. The resulting polynomial has all zeros on the (possibly infinite) circle of radius $\left | r \right |$ centered at $- r$, where $r = n/\sum {\lambda _j}$.