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The \((q, t)\)-Gaussian process. (English) Zbl 1264.46045
Summary: The \((q,t)\)-Fock space \(\mathcal F_{q,t}(\mathcal H)\), introduced in this paper, is a deformation of the \(q\)-Fock space of M. Bożejko and R. Speicher [Commun. Math. Phys. 137, No. 3, 519–531 (1991; Zbl 0722.60033)]. The corresponding creation and annihilation operators now satisfy the commutation relation \[ a_{q,t}(f)a_{q,t}(g)^{\ast} - qa_{q,t}(g)^{\ast}a_{q,t}(f)=\langle f,g\rangle _{\mathcal H}t^{N}, \] a defining relation of the Chakrabarti-Jagannathan deformed quantum oscillator algebra [R. Chakrabarti and R. Jagannathan, J. Phys. A, Math. Gen. 24, No. 13, L711-L718 (1991; Zbl 0735.17026)]. The moments of the deformed Gaussian element \(s_{q,t}(h):=a_{q,t}(h)+a_{q,t}(h)^{\ast}\) are encoded by the joint statistics of crossings and nestings in pair partitions. The \(q=0<t\) specialization yields a natural single-parameter deformation of the full Boltzmann Fock space of free probability, with the corresponding semicircular measure variously encoded via the Rogers-Ramanujan continued fraction, the \(t\)-Airy function, the \(t\)-Catalan numbers of L. Carlitz and J. Riordan [Duke Math. J. 31, 371–388 (1964; Zbl 0126.26301)], and the first-order statistics of the reduced Wigner process.

MSC:
46L54 Free probability and free operator algebras
46L53 Noncommutative probability and statistics
46L65 Quantizations, deformations for selfadjoint operator algebras
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