I know that if i have a Brownian motion $W_t$ the multiple Wiener integral

$\int_0^t \int_0^{\xi_1}...\int_0^{\xi_n} dW_{\xi_1}...dW_{\xi_n}$

can be expressed as $H_n(\int_0^t dW_s)$ where $H_n$ is the nth Hermite polynomial.

If now I have $m$ independent Brownian motion there is an analogous formula that allows us to express the integral $\int_0^t \int_0^{\xi_1}...\int_0^{\xi_n} dW_{\xi_1}^{j_1}...dW_{\xi_n}^{j_n}$

as a polynomial in the variables $\int_0^t dW_s^{j_1}...,\int_0^t dW_s^{j_n}$

I know that the polynomials $\{T_\alpha(\eta)=\prod_{\alpha_i} H_{\alpha_i}(\eta_i) \ \ \forall \ \ \alpha \text{ multi-index }\} $ where $\eta$ is a Gaussian random variable are widely used in the study of the Wiener chaos so i thought they could be a natural candidate, but i can't find any reference which express this result.