Wiener-Itô theorem in terms of Wick tensors (Q5937967)

Let us define the Hermite polynomial of degree \(n\) with parameter \(\sigma^2\) by \[ \colon x^n\colon_{\sigma^2}=(-\sigma^2)^n e^{x^2/2\sigma^2} D^n_x e^{-x^2/2\sigma^2} \] and define the Wick tensor as \(\:x^{\otimes n}\:= \sum_{k=0}^{[n/2]} {{n}\choose{2k}} (2k-1)!!(-1)^k \tau^{\otimes k} \widehat{\otimes}x^{\otimes(n-2k)}\), where \(\tau\) is the trace operator. Using above, the Wiener-Itô decomposition of an \(L^2\) functional \(\varphi\) is rewritten as \(\varphi(x) = \sum_{n=0}^{\infty}\langle \colon x^{\otimes n}\colon,u_n\rangle\). The multiple-Wiener integral of order \(n\) is given by \(I_n(f)=\langle\colon\cdot^{\otimes n}\colon,f\rangle\). And it holds \(I_{n+1}(f\otimes g)=I_1(f)I_n(g)-\sum_{k=1}^{n}I_{n-1}(f\otimes_k g)\).