Abelian Squares and Their Progenies

A polynomial $P \in \mathbb{C}[z_1, \ldots, z_d]$ is strongly $\mathbb{D}^d$-stable if $P$ has no zeroes in the closed unit polydisc $\overline{\mathbb{D}}^d.$ For such a polynomial define its spectral density function as $\mathcal{S}_P(\mathbf{z}) = \left(P(\mathbf{z})\overline{P(1/\overline{\mathbf{z}})}\right)^{-1}.$ An abelian square is a finite string of the form $ww'$ where $w'$ is a rearrangement of $w.$ We examine a polynomial-valued operator whose spectral density function's Fourier coefficients are all generating functions for combinatorial classes of constrained finite strings over a $d$-character alphabet. These classes generalize the notion of an abelian square, and their associated generating functions are the Fourier coefficients of one, and essentially only one, $L^2(\mathbb{T}^d)$-valued operator. Integral representations, divisibility properties, and recurrent and asymptotic behavior of the coefficients of these generating functions are given as consequences. Tools in the derivations of our asymptotic formulas include a version of Laplace's method for sums over lattice point translations due to Greenhill, Janson, and Ruci\'{n}ski, a version of stationary phase method for oscillatory integrals with complex phase due to Pemantle and Wilson, and various polynomial identities related to powers of modified Bessel functions of the first kind due to Moll and Vignat.