2011/11/28 by Natasha Blitvić, Blitvić, Natasha · 1 citation
Mathematics · Physics and Astronomy · #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Combinatorics (math.CO) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Operator Algebras (math.OA) #Quantum Algebra (math.QA) #Random Matrices and Applications #math-ph #math.CO #math.MP #math.OA #math.QA
paper · pdf · doi:10.48550/arxiv.1111.6565
The present version reverts to v2, by removing former Lemma 13 that contained an error
openalex publication_date 2011/11/28 · arxiv created 2012/03/21 · arxiv updated 2012/03/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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, \aq,t(h)^∗\h∈ H and \aq,t(h)\h∈ H, satifying the (q,t)-commutation relation aq,t(f)aq,t(g)^∗-q aq,t(g)^∗ aq,t(f)= H tN, for h,g∈ 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 sq,t(h):=aq,t(h)+aq,t(h)^∗, for h∈ H. The resulting refinement is particularly natural, as the moments of sq,t(h) are encoded by the joint statistics of crossings and nestings in pair partitions. Furthermore, the orthogonal polynomial sequence associated with the normalized (q,t)-Gaussian sq,t is that of the (q,t)-Hermite orthogonal polynomials, a deformation of the q-Hermite sequence that is given by the recurrence zHn(z;q,t)=Hn+1(z;q,t)+[n]q,tHn-1(z;q,t), with H0(z;q,t)=1, H1(z;q,t)=z, and [n]q,t=∑i=1n qi-1tn-i. The q=0