On the norm of the $q$-circular operator

The $q$-commutation relations, formulated in the setting of the $q$-Fock space of Bo\.zjeko and Speicher, interpolate between the classical commutation relations (CCR) and the classical anti-commutation relations (CAR) defined on the classical bosonic and fermionic Fock spaces, respectively. Interpreting the $q$-Fock space as an algebra of "random variables" exhibiting a specific commutativity structure, one can construct the so-called $q$-semicircular and $q$-circular operators acting as $q$-deformations of the classical Gaussian and complex Gaussian random variables, respectively. While the $q$-semicircular operator is generally well understood, many basic properties of the $q$-circular operator (in particular, a tractable expression for its norm) remain elusive. Inspired by the combinatorial approach to free probability, we revist the combinatorial formulations of these operators. We point out that a finite alternating-sum expression for $2n$-norm of the $q$-semicircular is available via generating functions of chord-crossing diagrams developed by Touchard in the 1950s and distilled by Riordan in 1974. Extending these norms as a function in $q$ onto the complex unit ball and taking the $n\to\infty$ limit, we recover the familiar expression for the norm of the $q$-semicircular and show that the convergence is uniform on the compact subsets of the unit ball. In contrast, the $2n$-norms of the $q$-circular are encoded by chord-crossing diagrams that are parity-reversing, which have not yet been characterized in the combinatorial literature. We derive certain combinatorial properties of these objects, including closed-form expressions for the number of such diagrams of any size with up to eleven crossings. These properties enable us to conclude that the $2n$-norms of the $q$-circular operator are significantly less well behaved than those of the $q$-semicircular operator.