An operator that relates to semi-meander polynomials via a two-sided $q$-Wick formula

Abstract

We consider the sequence $( Q_n )_{n=1}^{\infty}$ of semi-meander polynomials which are used in the enumeration of semi-meandric systems (a family of diagrams related to the classical stamp-folding problem). We show that for a fixed natural number $d$, the sequence $( Q_n (d) )_{n=1}^{\infty}$ appears as sequence of moments for a compactly supported probability measure $\nu_d$ on the real line. More generally, we consider a two-variable generalization $Q_n (t,u)$ of $Q_n(t)$, which is related to a natural concept of self-intersecting meandric system; the second variable of $Q_n (t,u)$ keeps track of the crossings of such a system (and one has, in particular, that $Q_n (t,0)$ is the original semi-meander polynomial $Q_n (t)$). We prove that for a fixed natural number $d$ and a fixed real number $q$ with $|q| < 1$, the sequence $( Q_n (d,q) )_{n=1}^{\infty}$ appears as sequence of moments for a compactly supported probability measure $\nu_{d:q}$ on the real line. The measure $\nu_{d;q}$ is found as scalar spectral measure for an operator $T_{d;q}$ constructed by using left and right creation/annihilation operators on a $q$-deformation of the full Fock space introduced by Bozejko and Speicher. The relevant calculations of moments for $T_{d;q}$ are made by using a two-sided version of a (previously studied in the one-sided case) $q$-Wick formula, which involves the number of crossings of a pair-partition.