Turán's inequalities
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In mathematics, Turán's inequalities are some inequalities for
Legendre polynomials In physical science and mathematics, Legendre polynomials (named after Adrien-Marie Legendre, who discovered them in 1782) are a system of complete and orthogonal polynomials, with a vast number of mathematical properties, and numerous applica ...
found by (and first published by ). There are many generalizations to other polynomials, often called Turán's inequalities, given by and other authors. If is the th Legendre polynomial, Turán's inequalities state that :\,\! P_n(x)^2 > P_(x)P_(x)\text-1 For ''H''''n'', the ''n''th Hermite polynomial, Turán's inequalities are :H_n(x)^2 - H_(x)H_(x)= (n-1)!\cdot \sum_^\fracH_i(x)^2>0 ~, whilst for
Chebyshev polynomials The Chebyshev polynomials are two sequences of polynomials related to the trigonometric functions, cosine and sine functions, notated as T_n(x) and U_n(x). They can be defined in several equivalent ways, one of which starts with trigonometric ...
they are :T_n(x)^2 - T_(x)T_(x)= 1-x^2>0 \text-1


See also

* Askey–Gasper inequality *
Sturm Chain In mathematics, the Sturm sequence of a univariate polynomial is a sequence of polynomials associated with and its derivative by a variant of Euclid's algorithm for polynomials. Sturm's theorem expresses the number of distinct real roots of ...


References

* * * Orthogonal polynomials Inequalities {{mathanalysis-stub