The function is defined as a solution to the differential equation

(d²G/dx²) + x(dG/dx) + λG = 0

Suppose G(x) is given by polynomial series

G(x) = Σ_m=0^∞c_mx^m

Then from the defining differential equation it follows that

(m+2)(m+1)c_m+2 + (m+λ)c_m = 0

The coefficient series will terminate (become all zeroes after a particular index) only if λ is equal to a negative integer, say n. The defining differential equation for G_n(x) is then

(d² GC/dx²) + x(d G_n/dx) −n G_n = 0

This differential equation is homogeneous so nothing is lost by taking the first nonzero coefficient to be unity. Two sets of G_n functions may be defined; one in which c_m is zero if m is even and another in which c_m is zero if m is odd. These correspond to the odd-evenness of the parameter n.

The Hermite polynomials obey a number of relationships. The G_n(x) functions obey the same ones. One of particular significance says that as n increases without bound G_n(x)² asymptotically approaches

2(2n/e)ⁿcos²(x(2n)^½ − nπ/2)(1 − ½x²/n))^½

This has the implication that the spatial average of the quantum theoretic probability distribution for a harmonic oscillator asymptotically approaches the classical time-spent probability distribution for a harmonic oscillator.

(To be continued.)

Appendix

The Hermite polynomial functions H_n(x) is relevant for determing the wavefunction for a harmonic oscillator, but always in the form

exp(−x²/2)H_n(x)

This suggests that it is this quantity that should be defined and investigated.

The Hermite polynomials are solutions to this differentia; equation

(e^−x²/2u')' + λe^−x²/2u = 0

Let e^−x²/2u(x)=w(x). Then

u(x) = e^x²/2w(x)
and
u'(x) = e^x²/2w'(x) + x·e^x²/2w(x)
e^−x²/2u'(x) = w'(x) + xw(x)

Thus the differential equation for Hermite polynomial is equivalent to

(w'(x) + xw(x))' + λw(x) = 0
and
w"(x) + xw'(x) + w(x) + λw(x) = 0
w"(x) + xw'(x) + (1 + λ)w(x) = 0

So the differential equation defining w(x) is

w"(x) + xw'(x) + μw(x) = 0

where μ is a real constant.