Hermite equation | Physics Library

The linear differential equation

d2f- df- dz2 − 2zdz + 2nf = 0,

in which n is a real constant, is called the Hermite equation. Its general solution is f := Af₁+Bf₂ with A and B arbitrary constants and the functions f₁ and f₂ presented as

f₁(z) := z + 2(1−n)
3! z³ + 22(1−n)(3−n)-
5! z⁵ + 23(1−-n)(3−n)(5−n)
7! z⁷ + ⋅⋅⋅ ,

f₂(z) := 1 + 2(−2n!) z² + 2
2-(−n)4(!2−-n) z⁴ + 3
2(−n)(2−6n!)(4−-n) z⁶ + ⋅⋅⋅

It’s easy to check that these power series satisfy the differential equation. The coefficients b_ν in both series obey the recurrence formula

2(ν− 2− n) bν = -----------bν−2. ν(nu − 1 )

Thus we have the radii of convergence

| | |bν−2| ν 1− 1∕ν R = lνim→∞ ||-b--|| = νli→m∞ 2⋅1-−-(n+2--)∕ν = ∞. ν

Therefore the series converge in the whole complex plane and define entire functions.

If the constant n is a non-negative integer, then one of f₁ and f₂ is simply a polynomial function. The polynomial solutions of the Hermite equation are usually normed so that the highest degree term is (2z)ⁿ and called the Hermite polynomials.

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