Turán's inequalities

In mathematics, Turán's inequalities are some inequalities for Legendre polynomials found by Paul Turán (1950) (and first published by Szegö (1948)). There are many generalizations to other polynomials, often called Turán's inequalities, given by (E. F. Beckenbach, W. Seidel & Otto Szász 1951) and other authors.

If P_n is the nth 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 H_n, the nth 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

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