Positivity of Turán determinants for orthogonal polynomials

Details Positivity of Turán determinants for orthogonal polynomials Positivity of Turán determinants for orthogonal polynomials

2007-10-17

arxiv.org/abs/0710.3389v1

in Harmonic Analysis and Hypergroups, (K.A. Ross et al., ed.) Delhi 1995, Birkhauser, Boston-Basel-Berlin, 1997, 165-182

Mathematics

Classical Analysis and ODEs

Scientific paper

The orthogonal polynomials $p_n$ satisfy Tur\'an's inequality if $p_n^2(x)-p_{n-1}(x)p_{n+1}(x)\ge 0$ for $n\ge 1$ and for all $x$ in the interval of orthogonality. We give general criteria for orthogonal polynomials to satisfy Tur\'an's inequality. This yields the known results for classical orthogonal polynomials as well as new results, for example, for the $q$--ultraspherical polynomials.

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