Finite sums of consecutive Hermite polynomials have real-rootedness governed by a coefficient polynomial: real roots of P force real roots of the sum, and non-real roots of P appear one-for-one in high-degree sums.
Deift, Orthogonal Polynomials and Random Matrices: a Riemann-Hilbert approach, in: Courant Lecture Notes in Mathematics, vol
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Zeros of linear combinations of Hermite polynomials
Finite sums of consecutive Hermite polynomials have real-rootedness governed by a coefficient polynomial: real roots of P force real roots of the sum, and non-real roots of P appear one-for-one in high-degree sums.