Turán's inequalities

In mathematics, Turán's inequalities are some inequalities for Legendre polynomials found by Pál Turán^[1] (and first published by Szegö (1948)^[2]). There are many generalizations to other polynomials, often called Turán's inequalities, given by (E. F. Beckenbach, W. Seidel & Otto Szász 1951)^[3] and other authors.

If ${\displaystyle P_{n}}$ is the ${\displaystyle n}$th Legendre polynomial, Turán's inequalities state that

${\displaystyle \,\!P_{n}(x)^{2}>P_{n-1}(x)P_{n+1}(x)\ {\text{for}}\ -1<x<1.}$

For ${\displaystyle H_{n}}$, the ${\displaystyle n}$th Hermite polynomial, Turán's inequalities are

${\displaystyle H_{n}(x)^{2}-H_{n-1}(x)H_{n+1}(x)=(n-1)!\cdot \sum _{i=0}^{n-1}{\frac {2^{n-i}}{i!}}H_{i}(x)^{2}>0,}$

whilst for Chebyshev polynomials they are

${\displaystyle T_{n}(x)^{2}-T_{n-1}(x)T_{n+1}(x)=1-x^{2}>0\ {\text{for}}\ -1<x<1.}$