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 detail 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 \(g(n)/f(n)^2 \rightarrow 0\) as \(n \rightarrow \infty \), 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.

对于为什么本福德的数字偏差定律在如此众多的不同领域和数据集中浮出水面，现在有许多理论上的解释。在简要回顾了其中一些内容之后，我们将详细讨论递归关系。由于它们是微分方程的离散类似物，并且可以模拟各种现实世界的现象，因此它们提供了用于测试Benfordness的系统的重要来源。先前的工作表明，具有恒定系数的固定深度递归是一些通常可以满足的技术假设的本福德模。我们简要回顾了该理论，然后证明了一些新结果，扩展到具有非恒定系数的线性递归关系的情况。我们证明，对于函数f和g的某些族，由形式的递归关系生成的序列\（a_ {n + 1} = f（n）a_n + g（n）a_ {n-1} \）是所有初始值的本福德。通过对系数进行参数化以获得较低程度的递归关系，然后转换为新的参数空间来进行证明。从那里，我们表明，合适的选择˚F和克其中˚F（Ñ）非降和\（G（N）/ F（N）^ 2 \ RIGHTARROW 0 \）作为\（N \ RIGHTARROW \ infty \） ，所述主项占主导地位，其行为等同于先前研究的均分布问题。我们还描述了进一步推广到具有非恒定系数的高阶递归关系和乘法递归关系的结果，以及当f和g是随机变量的值。