Arithmetic triangular structures in the transfer-matrix of finite Kronig-Penney models

This work provides a complete analytical characterization of the transfermatrix structure associated with the finite Kronig-Penney model consisting of one-dimensional arrays of Dirac delta potentials recently introduced by Figueroa et al. (2025). Although their study identified the emergence of specific transfer-matrix entries and related combinatorial coefficients, a rigorous derivation of the corresponding closed-form expressions has not yet been established. By expressing the Nth power of the unit-cell transfer matrix in terms of Chebyshev polynomials of the second kind, we obtain explicit closed-form representations for the global transmission and reflection amplitudes. The proposed formulation reveals a previously unrecognized structural correspondence between multiple quantum scattering processes, discrete convolutional patterns, and hypercomplex combinatorial structures.