For the Jacobi family with parameters $(0,\beta)$, $\beta\geq-1/3$, we prove a continuous comparison theorem for the associated weighted Jacobi--Radau functions. When two consecutive functions are parametrised by the same Pr\"ufer phase, the function of higher degree attains that phase closer to $\theta=0$ and has stri...
Let $z_{n,j}(\lambda)$ denote the positive zeros, in decreasing order, of the ultraspherical polynomial $C_n^\lambda$, $\lambda>-1/2$, with the reduced limiting interpretation at $\lambda=0$ specified below. Our principal result settles three higher-monotonicity questions of Gautschi: two as printed and the natural ope...
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
For $-1<\alpha\leq1/2$, let $J_\alpha$ be the Bessel function of the first kind and let $j_{\alpha,2}$ be its second positive zero. Define $\beta(\alpha)<\alpha+1$ by $$ \int_0^{j_{\alpha,2}}u^{-\beta(\alpha)}J_\alpha(u)\,du=0. $$ We prove that $\beta''(\alpha)>0$ for $-1<\alpha\leq1/2$, including the left second deriv...
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
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