How to prove that the sum of the squares of the roots of the nth Hermite polynomial is (n(n−1))/2?

Filipinacws 2022-08-13 Answered
How to prove that the sum of the squares of the roots of the nth Hermite polynomial is n ( n 1 ) 2 ?
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Answers (1)

Lisa Acevedo
Answered 2022-08-14 Author has 18 answers
Let us write
H n ( x ) = A n ( x x 1 ) ( x x n ) = (1) = A n ( x n e 1 ( x 1 , , x n ) x n 1 + e 2 ( x 1 , , x n ) x n 2 + p o l y n 3 ( x ) ) ,
where e k ( x 1 , , x n ) denote elementary symmetric polynomials:
e 1 ( x 1 , , x n ) = k = 1 n x k , e 2 ( x 1 , , x n ) = 1 i < j n n x i x j .
We want to find
(2) k = 1 n x k 2 = e 1 2 ( x 1 , , x n ) 2 e 2 ( x 1 , , x n ) ,
and therefore it will suffice to know the coefficients of x n , x n 1 and x n 2 in H n ( x ). But they can be determined from the series representations of Hermite polynomials:
(3) H n ( x ) = 2 n ( x n n ( n 2 ) 4 x n 2 + p o l y n 4 ( x ) ) .
Together with (1) and (2), this gives the result:
k = 1 n x k 2 = n ( n 1 ) 2 .
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