On polynomials satisfying a Turán type inequality

Csordas, George; Williamson, Jack

doi:10.1090/S0002-9939-1974-0338487-X

On polynomials satisfying a Turán type inequality
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by George Csordas and Jack Williamson

Proc. Amer. Math. Soc. 43 (1974), 367-372

DOI: https://doi.org/10.1090/S0002-9939-1974-0338487-X

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Abstract:

For Legendre polynomials ${P_n}(x)$, P. Turán has established the inequality \[ {\Delta _n}(x) = P_n^2(x) - {P_{n + 1}}(x){P_{n - 1}}(x) \geqq 0,\quad - 1 \leqq x \leqq 1,n \geqq 1,\] with equality only for $x = \pm 1$. This inequality has generated considerable interest, and analogous inequalities have been extended to various classes of polynomials: ultraspherical, Laguerre, Hermite, and a class of Jacobi polynomials. Our purpose here is to determine necessary and sufficient conditions for a general class of polynomials to satisfy a Turán type inequality and to characterize the generating functions of such a class.

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