The generating function for Hermite polynomials is a mathematical tool that, when expanded as a power series, yields the ratio of the Hermite polynomials and the factorial of the summation index as its coefficients.

It can also be used to derive the normalisation constant for the wavefunctions of the quantum harmonic oscillator, and is defined as

To prove the 2^{nd} equality of eq33, we take the partial derivative of with respect to to give . Substituting eq29 in this equation, we have

The solution to the differential equation is

When , eq34 becomes

Using eq19,

Using eq20,

Since the Maclaurin series expansion of is , we have , which completes the proof.

The generating function can be used to derive the ** Rodrigues formula** for Hermite polynomials. From 33,

Let . We have and . Therefore,

Substitute eq36 in eq35

When

From eq33

Substitute eq39 into eq38

Eq40 is the Rodrigues formula for Hermite polynomials. In other words, the Rodrigues formula for Hermite polynomials is a mathematical expression that provides a method to calculate any Hermite polynomial using differentiation.