Reference for Cauchy-type identity

Question

For polynomials $f,g \in \mathbb{C}[x_1,\dots,x_n]$, the following inner product appears frequently in the literature of harmonic polynomials: $$ \langle f,g \rangle = f(\partial/\partial x_1, \dots, \partial/\partial x_n) \cdot g(x_1,\dots,x_n) \mid_{x_1 = \dots = x_n = 0}. $$ In `Chains on Bruhat Order', the authors prove that two bases $\{f_\alpha\} $ and $\{g_\alpha\}$ (indexed by weak compositions) of $\mathbb{C}[x_1,x_2,\dots]$ are mutually orthogonal with respect to $\langle \cdot, \cdot \rangle$ if and only they satisfy the Cauchy-type identity $$ \prod_{i=1}^n e^{x_iy_i} = \sum_\alpha f_\alpha(x_1,\dots,x_n) \cdot g_\alpha(y_1,\dots,y_n). $$ They call the result "essentially well-known", with an attribution to an expository paper by Bergeron and Garsia "On certain spaces of harmonic polynomials" from 1993. Can anyone point me to an earlier reference for this identity?