The (q, t)-Gaussian process ✩

The (q, t)-Fock space Fq,t (H ), introduced in this paper, is a deformation of the q-Fock space of Bo˙zejko 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 H tN, 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. © 2012 Elsevier Inc. All rights reserved.