On the Relative Extrema of the Hermite Orthogonal Functions

1. The Hermite polynomials H_n (x) satisfy the recurrence relation

H_n+1 (x)=2xH_n(x)-2nH_n-1(x), n≥I,           (I.I)

H₀(x) = I, H₁(x) = 2x, H₂(x) = 4x²-2, H₃(x) = 8x³-12x.             (I.2)

We follow the notation of G. Szego [3, pp. 101-102]. We have

z"+(2n+1-x²) z = 0, Z = e^-x2/2 H_n(x) = z_n(x),            (I.3).