Binomial expansions of Jacobi-Stirling numbers and real-rootedness of Jacobi-Stirling descent polynomials

Abstract

In this paper, we first expand fixed diagonals of the Jacobi-Stirling numbers in a binomial basis. For the second kind, the expansion coefficients are polynomials in $z+1$ with nonnegative integer coefficients. We give a recurrence and a signed-partition interpretation for these coefficients. The same holds for the differences between corresponding unsigned first-kind and second-kind coefficients. We then prove that every nonzero nonnegative linear combination of the descent polynomials over Jacobi-Stirling permutations with a fixed number of deleted barred letters has only simple negative zeros when that number is one, two, or three. The same holds when exactly one or two barred letters are retained. Thus we verify five infinite families in a conjecture of Gessel, Lin and Zeng. Finally, using insertion operators, we find that every descent polynomial over Jacobi-Stirling permutations with a fixed number of deleted barred letters is top heavy and has an increasing left half.