Polynomials with real zeros and Polya frequency sequences

Abstract

Let $f(x)$ and $g(x)$ be two real polynomials whose leading coefficients have the same sign. Suppose that $f(x)$ and $g(x)$ have only real zeros and that $g$ interlaces $f$ or $g$ alternates left of $f$. We show that if $ad\ge bc$ then the polynomial $$(bx+a)f(x)+(dx+c)g(x)$$ has only real zeros. Applications are related to certain results of F.Brenti (Mem. Amer. Math. Soc. 413 (1989)) and transformations of P\'olya frequency sequences. More specifically, suppose that $A(n,k)$ are nonnegative numbers which satisfy the recurrence $$A(n,k)=(rn+sk+t)A(n-1,k-1)+(an+bk+c)A(n-1,k)$$ for $n\ge 1$ and $0\le k\le n$, where $A(n,k)=0$ unless $0\le k\le n$. We show that if $rb\ge as$ and $(r+s+t)b\ge (a+c)s$, then for each $n\ge 0$, $A(n,0),A(n,1),...,A(n,n)$ is a P\'olya frequency sequence. This gives a unified proof of the PF property of many well-known sequences including the binomial coefficients, the Stirling numbers of two kinds and the Eulerian numbers.