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V. Botta

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Preprint Aug 2026

A proof of Gautschi's conjecture on subrange Jacobi polynomials

Let $\pi_n$ be the monic polynomial of degree $n$ orthogonal on $[-c,c]$, $0<c\leq1$, with respect to the Jacobi weight $(1-x)^\alpha(1+x)^\beta$, where $-1<\alpha<\beta$. Gautschi conjectured that \[ \left[ \frac{\pi_n(-c)}{\pi_n(c)} \right]^2 \left(\frac{1-c}{1+c}\right)^{\beta-\alpha}<1. \] By his variation formula,...

V. Botta, K. Castillo, L. Tertuliano da Silva · 0 citations

Preprint Aug 2026

Degree-uniform regions for Gautschi's conjecture on subrange Jacobi polynomials

Let $\pi_n$ be the monic polynomial of degree $n$ orthogonal on $[-c,c]$, $0<c\leq 1$, with respect to the Jacobi weight $(1-x)^\alpha(1+x)^\beta$, where $-1<\alpha<\beta$. Gautschi conjectured that $$ \left[ \frac{\pi_n(-c)}{\pi_n(c)} \right]^2 \left(\frac{1-c}{1+c}\right)^{\beta-\alpha}<1. $$ By his variation formula...

V. Botta, K. Castillo, L. Tertuliano et al. · 0 citations