Quantum Decomposition Associated with the q-Deformed Lévy–Meixner White Noise

In this paper we start with a new detailed construction of the one-mode type q-Lévy-Meixner Fock space ${ℱ}_{LM}^{(q)}(\mathscr{H})$ which serves to obtain the quantum decomposition associated with the q-deformed Lévy-Meixner white noise processes. More precisely, based on the notion of quantum decomposition and the orthogonalization of polynomials of noncommutative q-Lévy-Meixner white noise $\omega (t):=b_t^{*}+c_2{b}_t+\beta b_t^{*}{b}_t^2+\gamma b_t^{*}{b}_t+c_{1}I$, we study the chaos property of the noncommutative L²-space with respect to the vacuum expectation τ. Next, we determine the distribution of the q-Lévy-Meixner operator J(χD) = ⟨ω, χD⟩ and as a consequence we give some useful properties of the q-Lévy-Meixner white noise process.