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Asymptotic monotonicity of the relative extrema of Jacobi polynomials. (English) Zbl 0819.33004

Let ${P}_{n}^{\left(\alpha ,\beta \right)}\left(x\right)$ denote the $n$-th Jacobi polynomial and let ${y}_{k,n}^{\left(\alpha ,\beta \right)}$ be the abscissas of the relative extrema of ${P}_{n}^{\left(\alpha ,\beta \right)}\left(x\right)/{P}_{n}^{\left(\alpha ,\beta \right)}\left(1\right)$ ordered by $-1={y}_{n,n}^{\left(\alpha ,\beta \right)}<{y}_{n-1,n}^{\left(\alpha ,\beta \right)}<\cdots <{y}_{1,n}^{\left(\alpha ,\beta \right)}<{y}_{0,n}^{\left(\alpha ,\beta \right)}$. Set

${\mu }_{k,n}\left(\alpha ,\beta \right)=\frac{{P}_{n}^{\left(\alpha ,\beta \right)}\left(x\right)}{{P}_{n}^{\left(\alpha ,\beta \right)}\left(1\right)},\phantom{\rule{2.em}{0ex}}k=1,\cdots ,n-1·$

A remarkable result obtained in the paper is the asymptotic representation, as $n\to \infty$ and for each fixed $k=1,2,\cdots$,

${\mu }_{k,n}\left(\alpha ,\beta \right)={\Gamma }\left(\alpha +1\right){\left(\frac{2}{{j}_{\alpha +1,k}}\right)}^{\alpha }\phantom{\rule{4pt}{0ex}}{J}_{\alpha }\left({j}_{\alpha +1,k}\right)\left[1+\frac{\alpha +3\beta +2}{24}\frac{{j}_{\alpha +1,k}^{2}}{{N}^{2}}+O\left({N}^{-4}\right)\right],$

where $N=n+\frac{1}{2}\left(\alpha +\beta +1\right)$ and ${j}_{\alpha +1,k}$ is the $k$-th positive zero of the Bessel function ${J}_{\alpha +1}\left(x\right)$. This representation, which corrects an earlier result of R. Cooper [Proc. Cambridge Phil. Soc. 46, 549- 554 (1950; Zbl 0038.223)], is not sufficient to prove a monotonicity property of the extrema ${\mu }_{k,n}\left(\alpha ,\beta \right)$ conjectured by Askey, i.e. that for $\alpha >\beta >-1/2$,

$|{\mu }_{k,n+1}\left(\alpha ,\beta \right)|<|{\mu }_{k,n}\left(\alpha ,\beta \right)|,\phantom{\rule{2.em}{0ex}}k=1,\cdots ,n\phantom{\rule{1.em}{0ex}}\text{and}\phantom{\rule{1.em}{0ex}}n=1,2,\cdots \phantom{\rule{4pt}{0ex}}·$

The authors are able to overcome this difficulty and show that Askey’s conjecture is true at least in the asymptotic sense. This is done by using another more powerful representation of ${\mu }_{k,n}\left(\alpha ,\beta \right)$, derived from a uniform asymptotic approximation of the Jacobi polynomial.

##### MSC:
 33C45 Orthogonal polynomials and functions of hypergeometric type 41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
##### Keywords:
Jacobi polynomial