Tura ́n–Hankel determinants of Stieltjes sequences: a pushforward proof and strict positivity refinements

For a Stieltjes moment sequence {an}n≥0 with representing measure μ on [0,∞), let Tr(n) = det[an+i+j]0≤i,j≤r be the Tur ́an–Hankel determinant of level r at shift n. We record two complementary results.

(1) A pushforward proof of the Stieltjes-moment property. {Tr(n)}n≥0 is itself a Stieltjes moment sequence, with explicit representing measure μr = W∗νr, the pushforward under the product map W(v0,...,vr) = vk of the Heine measure dνr = 1  (vj − k (r+1)! i<j vi)2 dμ⊗(r+1). The proof uses only Heine’s integral formula (1881), the change-of-variables formula for measure pushforward, and the elementary identity k vkn = (k vk)n. This is the same conclusion as Wang–Zhu (2016) for r = 1 and Zhu (2019) for general r (with an alternative proof by Park (2023)), reached without compound matrices, PSD characterisation, or lattice paths, and producing an explicit representing measure.

(2) Strict positivity refinements. Under |supp(μ)| = ∞, the pushforward measure μr has positive mass on (0,∞) and is not a single point mass; combined with Cauchy–Schwarz on L2(μr), this yields strict positivity Tr(n) > 0 and strict log-convexity Tr(n+1)2 < Tr(n)Tr(n+ 2) for all r,n ≥ 0, sharpening the non-strict Tr(n) ≥ 0 conclusion. The classical Desnanot– Jacobi recurrence Tr+1(n) Tr−1(n + 2) = Tr(n)Tr(n + 2) − Tr(n + 1)2 provides an alternative inductive route to strict positivity.

We illustrate both results on two benchmark Stieltjes sequences: the Riemann xi-derivatives an =ξ(2n)(21),verifiedat40-digitprecision,andtheHilbertmomentsan =1/(n+1),where the Barnes G-function asymptotic gives the analytic scaling ln Tr (0) ∼ −2(r + 1)2 ln 2.