Real-rootedness of Jacobi–Stirling descent polynomials

Prove the conjectured real-rootedness of the Jacobi–Stirling descent polynomials A_{k,i}(x), defined by summing the descent polynomials A_{k,S}(x) over all deletion sets S of cardinality i, for all admissible integers k and i.

Background

For a deletion set S of barred letters from the Jacobi–Stirling multiset, A_{k,S}(x) is the descent-generating polynomial of the resulting Jacobi–Stirling permutations. The polynomial A_{k,i}(x) is the sum of A_{k,S}(x) over all deletion sets S with |S|=i.

The paper notes that Brenti’s theorem establishes real-rootedness for each individual summand A_{k,S}(x), but that real-rootedness is not automatically preserved under summation. The authors prove the conjecture for the five infinite families i∈{1,2,3,k−2,k−1}; the quoted conjecture concerns the general real-rootedness assertion beyond those established families.

References

They observed that a theorem of Brenti gives the real-rootedness for each $A_{k,S}(x)$ and conjectured the same property for $A_{k,i}$ Conjecture~15.

— Binomial expansions of Jacobi-Stirling numbers and real-rootedness of Jacobi-Stirling descent polynomials  (2610.03111 - Ma et al., 2 Oct 2026) in Section 1, Introduction, paragraph preceding Theorem 2

By~eq:three-deletion, $A_{k-1,3} A_{k-2,0}$. Thus a sufficient condition for the real-rootedness of $A_{k,4}$ is $D_{3k-7}A_{k-1,4} A_{k-2,0}$. We do not establish this additional comparison here.

eq:three-deletion:

Ak,SAk−1,0.A_{k,S} A_{k-1,0}.

— Binomial expansions of Jacobi-Stirling numbers and real-rootedness of Jacobi-Stirling descent polynomials  (2610.03111 - Ma et al., 2 Oct 2026) in Remark following Corollary 4.5 in Section 4, “Proof of Theorem 2”