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Let $R_n^{(a,b)}=P_n^{(a,b)}/P_n^{(a,b)}(1)$ be the normalised Jacobi polynomials. For $\alpha\ge0$, put $\xi_0=-1$ and let $\xi_n$ be the largest zero of $R_n^{(\alpha+1,\alpha)}$ for $n\ge1$. We prove that $$ R_k^{(\alpha,\alpha)}(x) =\min_{n\ge0}R_n^{(\alpha,\alpha)}(x), \quad \xi_{k-1}\le x\le\xi_k, $$ for every $k...
For Jacobi polynomials with parameters greater than $-1/2$, and with the relative extrema enumerated from the endpoint $x=1$, the normalised modulus at the $k$th extremum of degree $n+1$ is proved to be strictly smaller than that at the $k$th extremum of degree $n$, for $1\leq k\leq n$. This proves the Szeg\H{o} conjec...
We settle two conjectures of Gautschi on the degree dependence of the zeros of the Jacobi polynomials $P_n^{(\alpha,\beta)}$, $\alpha,\beta>-1$, and obtain results substantially stronger than those conjectured. The conjectures stem from a line of questions originating in spherical cubature and hyperinterpolation. A Lio...
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