Department of

September 2014 October 2014 November 2014 Su Mo Tu We Th Fr Sa Su Mo Tu We Th Fr Sa Su Mo Tu We Th Fr Sa 1 2 3 4 5 6 1 2 3 4 1 7 8 9 10 11 12 13 5 6 7 8 9 10 11 2 3 4 5 6 7 8 14 15 16 17 18 19 20 12 13 14 15 16 17 18 9 10 11 12 13 14 15 21 22 23 24 25 26 27 19 20 21 22 23 24 25 16 17 18 19 20 21 22 28 29 30 26 27 28 29 30 31 23 24 25 26 27 28 29 30

Thursday, October 30, 2014

**Abstract:** For any fixed positive integer $n$, we study the zero distribution of a sequence of polynomials $H_{m}(z)$ satisfying the rational generating function \[ \sum_{m=0}^{\infty}H_{m}(z)t^{m}=\frac{1}{1+B(z)t+A(z)t^{n}} \] where $A(z)$ and $B(z)$ are any polynomials in $z$ with complex coefficients. We show that for all large $m$, the zeros of $H_{m}(z)$ which satisfy $A(z)\ne0$ lie on the fixed real algebraic curve given by \[ \Im\frac{B^{n}(z)}{A(z)}=0\qquad\mbox{and}\qquad0\le(-1)^{n}\Re\frac{B^{n}(z)}{A(z)}\le\frac{n^{n}}{(n-1)^{n-1}} \] and are dense there as $m\rightarrow\infty$.