Turán's inequalities explained

In mathematics, Turán's inequalities are some inequalities for Legendre polynomials found by (and first published by). There are many generalizations to other polynomials, often called Turán's inequalities, given by and other authors.

If

P _{n} is the n th Legendre polynomial, Turán's inequalities state thatP ^{2} _{n(x)} > P _{n-1} (x)P _{n+1} (x) for -1 <x< 1 .

For

H _{n} , the n th Hermite polynomial, Turán's inequalities areH ^{2} _{n(x)} - H _{n-1} (x)H _{n+1} (x) = (n -1 )! ⋅ \sum ^{n -1} _{i =0} (2 ^{n-i})/(i!) H ^{2 > 0} _{i(x)} ,

whilst for Chebyshev polynomials they are

T ^{2} _{n(x)} - T _{n-1} (x)T _{n+1} (x) = 1- x ^{2>0} for -1 <x< 1 .

See also

• Askey–Gasper inequality
• Sturm Chain