From Gegenbauer to Hermite | M.E. Irizarry-Gelpí

The generating function for Gegenbauer uni-variate polynomials $C_{n}{}^{\lambda}(x)$ is

$$ \left( \frac{1}{1 - 2xr + r^{2}} \right)^{\lambda} = \sum_{n = 0}^{\infty} r^{n} C_{n}{}^{\lambda}(x). $$

The exponential generating function for Hermite uni-variate polynomials $H_{n}(x)$ is

$$ \exp{\left(2xr - r^{2} \right)} = \sum_{n = 0}^{\infty} \frac{r^{n}}{\Gamma(n+1)} H_{n}(x). $$

Recall the definition of the exponential function as a limit:

$$ \exp{(x)} = \lim_{\lambda \rightarrow \infty} \left(1 + \frac{x}{\lambda} \right)^{\lambda}. $$

You can use this relation to find an expression for Hermite uni-variate polynomials as a limit of Gegenbauer uni-variate polynomials.

First you perform a scale transformation:

$$ r \longrightarrow \frac{r}{\sqrt{\lambda}}, \qquad x \longrightarrow \frac{x}{\sqrt{\lambda}}. $$

Then you get

$$ \left(1 + \frac{r^{2} - 2xr}{\lambda} \right)^{-\lambda} = \sum_{n = 0}^{\infty} r^{n} \left( \frac{1}{\sqrt{\lambda}} \right)^{n} C_{n}{}^{\lambda}\left( \frac{x}{\sqrt{\lambda}} \right). $$

Taking the limit $\lambda \rightarrow \infty$ gives

$$ \exp{\left(2xr - r^{2} \right)} = \sum_{n = 0}^{\infty} \frac{r^{n}}{\Gamma(n+1)} \left[ \lim_{\lambda \rightarrow \infty} \Gamma(n+1) \left( \frac{1}{\sqrt{\lambda}} \right)^{n} C_{n}{}^{\lambda}\left( \frac{x}{\sqrt{\lambda}} \right) \right]. $$

Thus,

$$ H_{n}(x) = \lim_{\lambda \rightarrow \infty} \Gamma(n+1) \left( \frac{1}{\sqrt{\lambda}} \right)^{n} C_{n}{}^{\lambda}\left( \frac{x}{\sqrt{\lambda}} \right). $$

A similar trick holds for the multi-variate polynomials.