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The (q,t)-Gaussian Process

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<mark>Journal publication date</mark>15/11/2012
<mark>Journal</mark>Journal of Functional Analysis
Issue number10
Number of pages36
Pages (from-to)3270-3305
Publication StatusPublished
Early online date17/09/12
<mark>Original language</mark>English


The (q,t) -Fock space Fq,t(H) , introduced in this paper, is a deformation of the q-Fock space of Bożejko and Speicher. The corresponding creation and annihilation operators now satisfy the commutation relation aq,t(f)aq,t(g)⁎−qaq,t(g)⁎aq,t(f)=⟨f,g⟩HtN, Turn MathJax off a defining relation of the Chakrabarti–Jagannathan deformed quantum oscillator algebra. The moments of the deformed Gaussian element sq,t(h):=aq,t(h)+aq,t(h)⁎ 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 Carlitz–Riordan, and the first-order statistics of the reduced Wigner process.