Unveiling arithmetic triangular structures in the Kronig--Penney model

The one-dimensional quantum scattering of a particle with positive energy $E$ interacting with a strongly localized potential---modeled by a finite array of $N$ Dirac delta distributions---constitutes the finite Kronig--Penney model, a paradigmatic system in condensed matter physics. The dynamics are governed by the stationary Schr\"odinger equation $$ -\frac{\hbar^2}{2m}\psi''(x)+\lambda\sum_{n=1}^{N}\delta(x-nL)\psi(x) = E\psi(x), $$ where $L$ denotes the lattice spacing and $\lambda$ the coupling constant. As a quantum wave propagates through this array, multiple reflection and transmission events interfere coherently, defining the system's global scattering properties. The transfer-matrix formalism provides the systematic framework to analyze this process, as it relates the wave amplitudes across the array through the ordered product of elementary scattering matrices. Recent investigations into this structure have uncovered a surprising connection to integer sequences, [3]. Specifically, it has been observed that the $(1,1)$-entry of the principal transfer matrix $$ M^N(\alpha,\beta) = \begin{pmatrix} \alpha & -K\\ K^{-1} & \beta \end{pmatrix}^{N}, \qquad K\neq0, $$ generates non-symmetric triangular arrays of integers reminiscent of those found in hypergeometric function theory and generalized Appell polynomials, [1]-[2]. While these patterns are clearly related to several sequences listed in the \textit{On-Line Encyclopedia of Integer Sequences} (OEIS), their emergence has remained largely empirical. To date, the underlying mathematical mechanism responsible for these integer structures has not been rigorously characterized, leaving them as a fascinating, yet unexplained, phenomenon. In this talk, we aim to bridge this gap. We move beyond observation to reveal the hidden architecture governing the $(1,1)$-entry of the $N$th power of the transfer matrix. We will demonstrate how these coefficients emerge from specific discrete convolution structures, transforming what was once an empirical curiosity into a clear combinatorial mechanism. Our derivation reveals a previously unrecognized structural correspondence between multiple quantum scattering processes, discrete convolution patterns, and hypercomplex combinatorial structures. By providing a rigorous analytical explanation for these patterns, we offer a new perspective on the organization of scattering dynamics and the profound interplay between transfer-matrix methods, special functions, and arithmetic combinatorics.