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I present a partly pedagogic discussion of the Gel'fand-Yaglom formula forthe functional determinant of a one-dimensional, second order differenceoperator, in the simplest settings. The formula is a textbook one in discreteSturm-Liouville theory and orthogonal polynomials. A two by two matrix approachis developed and applied to Robin boundary conditions. Euler-Rayleigh sums ofeigenvalues are computed. A delta potential is introduced as a simple,non-trivial example and extended, in an appendix, to the general case. Thecontinuum limit is considered in a non--rigorous way and a rough comparisonwith zeta regularised values is made. Vacuum energies are also considered inthe free case. Chebyshev polynomials act as free propagators and theirproperties are developed using the two-matrix formulation, which has someadvantages and appears to be novel. A trace formula, rather than a determinantone, is derived for the Gel'fand-Yaglom function.

Discrete Determinants and the Gel'fand-Yaglom formula