The $(q,t)$-Gaussian Process

TL;DR

This paper introduces a two-parameter deformation of Fock spaces, leading to a new class of Gaussian processes with moments linked to crossings and nestings, and connects to various special functions and identities.

Contribution

The paper develops the $(q,t)$-Fock space and Gaussian process, extending previous $q$-deformations and linking moments to combinatorial structures and special functions.

Findings

Defined the $(q,t)$-Fock space and operators.

Connected moments to crossings and nestings in partitions.

Linked the probability measure to Rogers-Ramanujan identities and related functions.

Abstract

We introduce a two-parameter deformation of the classical Bosonic, Fermionic, and Boltzmann Fock spaces that is a refinement of the $q$-Fock space of [BS91]. Starting with a real, separable Hilbert space $H$, we construct the $(q,t)$-Fock space and the corresponding creation and annihilation operators, $\{a_{q,t}(h)^\ast\}_{h\in H}$ and $\{a_{q,t}(h)\}_{h\in H}$, satifying the $(q,t)$-commutation relation $a_{q,t}(f)a_{q,t}(g)^\ast-q \,a_{q,t}(g)^\ast a_{q,t}(f)= <f,g>_{_H}\, t^{N},$ for $h,g\in H$, with $N$ denoting the number operator. Interpreting the bounded linear operators on the $(q,t)$-Fock space as non-commutative random variables, the analogue of the Gaussian random variable is given by the deformed field operator $s_{q,t}(h):=a_{q,t}(h)+a_{q,t}(h)^\ast$, for $h\in H$. The resulting refinement is particularly natural, as the moments of $s_{q,t}(h)$ are encoded by the joint statistics of crossings \emph{and nestings} in pair partitions. Furthermore, the orthogonal polynomial sequence associated with the normalized $(q,t)$-Gaussian $s_{q,t}$ is that of the $(q,t)$-Hermite orthogonal polynomials, a deformation of the $q$-Hermite sequence that is given by the recurrence $zH_n(z;q,t)=H_{n+1}(z;q,t)+[n]_{q,t}H_{n-1}(z;q,t),$ with $H_0(z;q,t)=1$, $H_1(z;q,t)=z$, and $[n]_{q,t}=\sum_{i=1}^n q^{i-1}t^{n-i}$. The $q=0<t$ specialization yields a new single-parameter deformation of the full Boltzmann Fock space of free probability. The probability measure associated with the corresponding deformed semicircular operator turns out to be encoded, in various forms, via the Rogers-Ramanujan continued fraction, the Rogers-Ramanujan identities, the $t$-Airy function, the $t$-Catalan numbers of Carlitz-Riordan, and the first-order statistics of the reduced Wigner process.