Meixner class of orthogonal polynomials of a non-commutative monotone Levy noise

Let $(X_t{)}_{t\ge 0}$ denote a non-commutative monotone L'evy process. Let $\omega =(\omega (t){)}_{t\ge 0}$ denote the corresponding monotone L'evy noise… A continuous polynomial of $\omega$ is an element of the corresponding non-commutative $L^2$-space $L^2(\tau )$ that has the form $\sum_{i=0}^n\langle \omega^{\otimes i},f^{(i)}\rangle$,where$f^{(i)}\in C_0(\mathbb R_+^i)$. We denote by $\mathbf{C}\mathbf{P}$ the space of all continuous polynomials of $\omega$. For $f^{(n)}\in C_0({\mathbb{R}}_{+}^n)$, the orthogonal polynomial $\langle P^{(n)}(\omega),f^{(n)}\rangle$ is defined as the orthogonal projection of the monomial $⟨{\omega }^{\otimes n},f^{(n)}⟩$ onto the subspace of $L^2(\tau )$ that is orthogonal to all continuous polynomials of $\omega$ of order $\le n-1$. We denote by $\mathbf{O}\mathbf{C}\mathbf{P}$ the linear span of the orthogonal polynomials. Each orthogonal polynomial $\langle P^{(n)}(\omega),f^{(n)}\rangle$dependsonlyontherestrictionofthefunction$f^{(n)}$totheset${(t_1,\dots,t_n)\in\mathbb R_+^n\mid t_1\ge t_2\ge\dots\ge t_n}$. The orthogonal polynomials allow us to construct a unitary operator $J:L^2(\tau )\to \mathbb{F}$, where $\mathbb{F}$ is an extended monotone Fock space. Thus, we may think of the monotone noise $\omega$ as a distribution of linear operators acting in $\mathbb{F}$. We say that the orthogonal polynomials belong to the Meixner class if $\mathbf{C}\mathbf{P}=\mathbf{O}\mathbf{C}\mathbf{P}$. We prove that each system of orthogonal polynomials from the Meixner class is characterized by two parameters: $\lambda\in\mathbb R$and$\eta\ge0$. In this case, the monotone L'evy noise has the representation $\omega (t)={\mathrm{\partial }}_t^{\mathrm{†}}+\lambda {\mathrm{\partial }}_t^{\mathrm{†}}{\mathrm{\partial }}_t+{\mathrm{\partial }}_t+\eta {\mathrm{\partial }}_t^{\mathrm{†}}{\mathrm{\partial }}_t{\mathrm{\partial }}_t$. Here, ${\mathrm{\partial }}_t^{\mathrm{†}}$ and ${\mathrm{\partial }}_t$ are the (formal) creation and annihilation operators at $t\in {\mathbb{R}}_{+}$ acting in $\mathbb{F}$.