  There are now many theoretical explanations for why Benford's law of digit
bias surfaces in so many diverse fields and data sets. After briefly reviewing
some of these, we discuss in depth recurrence relations. As these are discrete
analogues of differential equations and model a variety of real world
phenomena, they provide an important source of systems to test for Benfordness.
Previous work showed that fixed depth recurrences with constant coefficients
are Benford modulo some technical assumptions which are usually met; we briefly
review that theory and then prove some new results extending to the case of
linear recurrence relations with non-constant coefficients. We prove that, for
certain families of functions $f$ and $g$, a sequence generated by a recurrence
relation of the form $a_{n+1} = f(n)a_n + g(n)a_{n-1}$ is Benford for all
initial values. The proof proceeds by parameterizing the coefficients to obtain
a recurrence relation of lower degree, and then converting to a new parameter
space. From there we show that for suitable choices of $f$ and $g$ where $f(n)$
is nondecreasing and $g(n)/f(n)^2 \to 0$ as $n \to \infty$, the main term
dominates and the behavior is equivalent to equidistribution problems
previously studied. We also describe the results of generalizing further to
higher-degree recurrence relations and multiplicative recurrence relations with
non-constant coefficients, as well as the important case when $f$ and $g$ are
values of random variables.
