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

Question

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

Lisa Acevedo · Accepted Answer

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            &amp;lt;            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    .

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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