Interlacing on the Unit Circle via Coefficientwise Reciprocals

Source abstract

Let $f(z)=\sum_{k=0}^{n}a_kz^k$ be a polynomial with positive coefficients, and define its coefficientwise reciprocal by $f^{\#}(z)=\sum_{k=0}^{n}\frac{z^k}{a_k}.$ It is known that if $f$ is palindromic and has only negative real zeros, then all zeros of $f^{\#}$ lie on the unit circle. We prove that if $p$ and $q$ are palindromic polynomials of degrees $n$ and $n+1$, respectively, with only negative real zeros, then their coefficientwise reciprocals have only simple zeros, and $p^{\#}$ strictly interlaces $q^{\#}$ on the unit circle. Our proof is based on finite Blaschke products and the comparison of their boundary phases. As an immediate consequence, we obtain strict interlacing for the reciprocal binomial, reciprocal Eulerian, and reciprocal Narayana polynomials. By combining a sign change in the $γ$-coefficients with coefficientwise reciprocation, we further construct strictly interlacing families from Rogers--Szegő, Poupard, and Kreweras-related polynomials.