Gautschi's conjecture on subrange Jacobi polynomials in the negative wedge: an endpoint obstruction to the first-crossing method and an effective reduction to finitely many degrees
Abstract
Let πₙ be the monic polynomial of degree n orthogonal on [-c,c], 0<c ≤ 1, with respect to the Jacobi weight (1-x)^α(1+x)^β, -1<α<β. Gautschi conjectured that [πₙ(-c)/πₙ(c)]² ((1-c)/(1+c))^(β-α)<1 for every n ≥ 1. Botta, Castillo and Tertuliano da Silva (arXiv:2608.05963) proved it, uniformly in the degree, throughout β ≥ 0, and in the remaining negative wedge, written α=-r-λ, β=-r+λ, whenever c² ≤ 3/(3+r); their method is a first-crossing argument whose pivot is the root-sum inequality Σ_N<λ, and their Remark 5.2 proposes to reach the rest of the wedge through the estimate C_(N,a)>λ((N+r)a²-N), where C_(N,a) is defined by an exact Ward identity. We show that the proposed estimate is false at the endpoint a=1, for every degree N ≥ 1 and every point of the negative wedge: with Σ_N(1) the classical sum of the zeros of the Jacobi polynomial, C_(N,1)=0 while the estimate requires C_(N,1)>λ r>0; equivalently, the root-sum inequality itself fails at c=1 throughout the negative sector. We also run the authors' own induction from a shifted start and obtain an effective statement: if the conjecture holds at (α,β) in every degree 1 ≤ n ≤ n₀ and on every subrange [-a,a] with a ≤ c, and rc² ≤ (n₀+2)(1-c²), then it holds in every degree; at n₀=1 this is exactly the published condition c² ≤ 3/(3+r), and at every fixed c<1 it reduces the degree-uniform conjecture to the explicit finite block 2 ≤ n ≤ ⌈ rc²/(1-c²)⌉-2, where the source has only a non-effective finiteness statement. A numerical scan of 219,816 cells of the open region, reported as a measurement, finds no counterexample. The arithmetic of every numbered statement not labelled otherwise is machine-checked in Lean 4; the analytic inputs are the source's lemmas and enter as hypotheses.
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Claim ledger
Stated results
GW1known2026-09-03
The sum of the zeros of the degree-n Jacobi polynomial, telescoped from the classical recurrence coefficients: Sigmaₙ = n(beta-alpha)/(2n+alpha+beta) for -2 < alpha+beta
GW2routine2026-09-03
At c = 1 the sum of the zeros lies above / at / below lambda = (beta-alpha)/2 according as alpha+beta is negative / zero / positive – the root-sum pivot of the source's method fails exactly in the negative sector
GW3routine2026-09-03
The Pearson quantity vanishes identically at the endpoint: lambda - Sigmaₙ - (n+nu) bₙ = 0 for every degree at c = 1
GW4candidate2026-09-03
The quantitative estimate proposed in Remark 5.2 of arXiv:2608.05963v1 is FALSE at the endpoint: C_(N,1) = 0 while the estimate requires C_(N,1) > lambda r > 0, for every degree N >= 1 and every point of the negative wedge
GW5routine2026-09-03
Non-vacuity control: the negative wedge is inhabited and the hypotheses of the endpoint theorems are satisfiable – (alpha,beta) = (-3/5,-1/5) gives Sigma₂ = 1/4 > 1/5 = lambda
GW6routine2026-09-03
The Ward bound to root-sum step, made exact: (N - r a²) Sigma < N lambda a² together with a²(N+r) <= N gives Sigma < lambda; the threshold solved for N; and N/(N+r) increasing in N
GW7routine2026-09-03
At a zero of dₙ the Pearson identity makes Sigmaₙ < lambda and Sigmaₙ₊₁ < lambda EQUIVALENT (for r < n); and positivity of dₙ propagates Sigmaₙ <= lambda to Sigmaₙ₊₁ < lambda
GW8routine2026-09-03
The root-sum inequality Sigma_N < lambda is EQUIVALENT to Remark 5.2's estimate C_(N,a) > lambda((N+r)a² - N), not merely implied by it
GW9candidate2026-09-03
An EFFECTIVE reduction of the degree-uniform conjecture in the negative wedge to the finite block of degrees 2 <= n <= ceil(r c²/(1-c²)) - 2: if the conjecture holds for all 1 <= n <= n0 and all a in (0,c], and r c² <= (n0+2)(1-c²), then it holds at every degree
GW10routine2026-09-03
Each further initial degree strictly widens the degree-uniform threshold from (n0+2)/(n0+2+r) to (n0+3)/(n0+3+r); 3/(3+r) < 4/(4+r) (so degree two alone would move the source's c <= sqrt(3)/2 to c <= 2/sqrt(5)); and the thresholds exhaust (0,1)
GW11routine2026-09-03
Worked instance: at r = 9/10, c = 19/20 (a point of the open region O_c, since c² = 361/400 > 3/(3+r) = 10/13) the reduction needs n0 = 7 and no less – degrees 2..7 are the whole remaining content of the conjecture there
GW12routine2026-09-03
Negative controls: the Ward-to-root-sum step is false without its threshold hypothesis (N=3, r=1/2, a=1, lambda=1, Sigma=11/10), and the crossing equivalence is false without r < n (r=2, n=1)
GW13measurement2026-09-03
MEASUREMENT: Gautschi's conjecture holds at all 219,816 cells of an 18,318-point scan of the OPEN region O_c at degrees 1..12 (minimum dₙ = 4.04e-6), with the Pearson and endpoint routes to dₙ agreeing to 1e-7 relative at every cell
This ledger entry is reported in prose and is not bound to a Lean theorem.GW14measurement2026-09-03
MEASUREMENT: the root-sum inequality Sigma_N < lambda holds only up to a measured threshold a*_N < 1 – a*₂ in 0.9907..0.9994 and a*₃ in 0.9964..0.9998 across the wedge, against the source's degree-uniform sqrt(3/(3+r)) in 0.871..0.968
This ledger entry is reported in prose and is not bound to a Lean theorem.GW15prose2026-09-03
PROSE: continuity of the moments in c at c = 1 upgrades GW4 from the endpoint to a punctured neighbourhood – for every wedge point and every degree N there is c_N < 1 with Sigma_N(c) > lambda for c in (c_N, 1)
This ledger entry is reported in prose and is not bound to a Lean theorem.Provenance
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- Source context
- For -1 < α < β and 0 < c ≤ 1, let πₙ = πₙ^((α,β))(·; c) be the monic polynomial of degree n orthogonal on [-c, c] with respect to the Jacobi weight w(x) = (1-x)^α (1+x)^β. Gautschi's conjecture (W. Gautschi, *On the zeros of subrange Jacobi polynomials*, Numer. Algorithms 79 (2018) 759–768) asserts, for every n ≥ 1 and every admissible (α, β, c),
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