Interlacing of the roots of the filter-integral polynomials of Amdeberhan, Duncan, Moll and Sharma

Abstract

In their study of filter integrals for orthogonal polynomials, Amdeberhan, Duncan, Moll and Sharma introduce the polynomials Xₙ(a) = 2²ⁿ⁻¹(a+1/2)ₙ-C(2n-1, n-1)(a)ₙ, prove that all n zeros of Xₙ are real and negative — their argument placing one at a=-n and one in each interval (-(j+1),-j) — and close their paper with a Problem: do the roots of Zₙ=Xₙ(a)/(a+n) interlace those of Zₙ₋₁? We prove that they do, for every n ≥ 2. The one ingredient absent from the source is an evaluation at half-integers: at a=-(j+1/2) the shifted Pochhammer (a+1/2)ₙ vanishes exactly as the unshifted (a)ₙ vanishes at a=-j, and the two evaluations carry the same sign (-1)ʲ. Consequently every root of Xₙ lies in the left half (-(j+1),-(j+1/2)) of its unit interval; the sign of (a)ₙ₋₁ at a root of Xₙ₋₁ is then determined, and interlacing follows from a two-term transfer identity in three lines. We also prove that n | Xₙ(a) for every a ∈ ℤ and every n, a question raised in the first arXiv version of the source and absent from the published version, from whose Property 6.16 only the prime case follows. Every statement below is machine-checked in Lean 4.

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