THC Science

In mathematics, Turán's inequalities are some inequalities for Legendre polynomials found by Pál 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 $n$ th Legendre polynomial, Turán's inequalities state that

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

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

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

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

References[edit]

Beckenbach, E. F.; Seidel, W.; Szász, Otto (1951), "Recurrent determinants of Legendre and of ultraspherical polynomials", Duke Math. J., 18: 1–10, doi:10.1215/S0012-7094-51-01801-7, MR 0040487
Szegö, G. (1948), "On an inequality of P. Turán concerning Legendre polynomials", Bull. Amer. Math. Soc., 54 (4): 401–405, doi:10.1090/S0002-9904-1948-09017-6, MR 0023954
Turán, Paul (1950), "On the zeros of the polynomials of Legendre", Časopis Pěst. Mat. Fys., 75 (3): 113–122, doi:10.21136/CPMF.1950.123879, MR 0041284

THC Science

Bringing Science to the Cannabis Conversation!

Legality of Cannabis by U.S. Jurisdiction

See also[edit]

References[edit]