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

Let $(X_t)_{t\ge0}$ denote a non-commutative monotone Lévy process. Let $ω=(ω(t))_{t\ge0}$ denote the corresponding monotone Lévy noise.. A continuous polynomial of $ω$ is an element of the corresponding non-commutative $L^2$-space $L^2(τ)$ that has the form $\sum_{i=0}^n\langle ω^{\otimes i},f^{(i)}\rangle$, where $f^{(i)}\in C_0(\mathbb R_+^i)$. We denote by $\mathbf{CP}$ the space of all continuous polynomials of $ω$. For $f^{(n)}\in C_0(\mathbb R_+^n)$, the orthogonal polynomial $\langle P^{(n)}(ω),f^{(n)}\rangle$ is defined as the orthogonal projection of the monomial $\langleω^{\otimes n},f^{(n)}\rangle$ onto the subspace of $L^2(τ)$ that is orthogonal to all continuous polynomials of $ω$ of order $\le n-1$. We denote by $\mathbf{OCP}$ the linear span of the orthogonal polynomials. Each orthogonal polynomial $\langle P^{(n)}(ω),f^{(n)}\rangle$ depends only on the restriction of the function $f^{(n)}$ to the set $\{(t_1,\dots,t_n)\in\mathbb R_+^n\mid t_1\ge t_2\ge\dots\ge t_n\}$. The orthogonal polynomials allow us to construct a unitary operator $J:L^2(τ)\to\mathbb F$, where $\mathbb F$ is an extended monotone Fock space. Thus, we may think of the monotone noise $ω$ as a distribution of linear operators acting in $\mathbb F$. We say that the orthogonal polynomials belong to the Meixner class if $\mathbf{CP}=\mathbf{OCP}$. We prove that each system of orthogonal polynomials from the Meixner class is characterized by two parameters: $λ\in\mathbb R$ and $η\ge0$. In this case, the monotone Lévy noise has the representation $ω(t)=\partial_t^†+λ\partial_t^†\partial_t+\partial_t+η\partial_t^†\partial_t\partial_t$. Here, $\partial_t^†$ and $\partial_t$ are the (formal) creation and annihilation operators at $t\in\mathbb R_+$ acting in $\mathbb F$.