Zeros distribution and interlacing property for certain polynomial sequences

In this article, we first prove that the Hankel determinant of order three of the polynomial sequence { P n ( x ) = & sum; k >= 0 P ( n , k ) x k } n >= 0 {\left\{{P}_{n}\left(x)={\sum }_{k\ge 0}P\left(n,k){x}<^>{k}\right\}}_{n\ge 0} is weakly (Hurwitz) stable, where P ( n , k ) P\left(n,k) satisfies the recurrence relation P ( n , k ) = ( a 1 n + a 2 ) P ( n - 1 , k ) + ( b 1 n + b 2 ) P ( n - 1 , k - 1 ) , P\left(n,k)=\left({a}_{1}n+{a}_{2})P\left(n-1,k)+\left({b}_{1}n+{b}_{2})P\left(n-1,k-1), with P ( n , k ) = 0 P\left(n,k)=0 wherever k is not an element of { 0 , 1 , & mldr; , n } . k\notin \left\{0,1,\ldots ,n\right\}. The stability of a polynomial is closely associated with the interlacing property, which is based on the Hermite-Biehler theorem. We also show the interlacing property of the polynomial sequence ( U n ( x ) ) n >= 0 , {\left({U}_{n}\left(x))}_{n\ge 0}, which satisfies the following recurrence relation: U n ( x ) = ( alpha n x + beta n ) U n - 1 ( x ) + ( u n x 2 + v n x ) U n - 1 ' ( x ) {U}_{n}\left(x)=\left({\alpha }_{n}x+{\beta }_{n}){U}_{n-1}\left(x)+\left({u}_{n}{x}<^>{2}+{v}_{n}x){U}_{n-1}<^>{<^>{\prime} }\left(x) based on the Hermite-Biehler theorem. As applications, we obtain the weak (Hurwitz) stability of the Hankel determinant of order three for the row polynomials of the (unsigned) Stirling numbers of the first kind, the Whitney numbers of the first kind, and show the interlacing property of Eulerian polynomials, Bell polynomials, and Dowling polynomials.