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# Recurrence relations and Benford’s law

## Abstract

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.

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## Notes

1. 1.

If $$\log _{10} 2$$ equals p/q then $$2 = 10^{p/q}$$, so $$2^{q-p} = 5^p$$ and thus $$p = q-p = 0$$, which is impossible.

2. 2.

We are using big-Oh notation: $$f(x) = O(g(x))$$ at infinity if there exists an $$x_0$$ and a $$C>0$$ such that for all $$x \ge x_0$$ we have $$|f(x)| \le C g(x)$$; it is big-Oh at zero if instead it is true for all $$x \le x_0$$.

3. 3.

Unlike our case, for roots of polynomials this process breaks down for degree five and higher.

4. 4.

These functions are not uniquely determined by fg; however, they are completely determined by a choice of $$\lambda (1)$$.

5. 5.

As in Lemma 3.4, these functions are not unique, but are completely determined by $$\lambda (1), \lambda (2)$$.

6. 6.

Generalizing the recurrence relation of Fibonacci k-step numbers, the authors proved the recursive sequence $$x_n = \prod _{i=1}^{k} x_{n-i}$$ to be Benford. Each term $$x_n$$ ($$n>k$$) is the product of powers of $$x_1, \cdots , x_k$$ with Fibonacci k-step numbers as the exponents.

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## Acknowledgements

Much of the analysis was done during the 2018 Williams College SMALL REU Program, and we are grateful to our colleagues there, as well as to participants of the Conference on Benford’s Law Application in Stresa, Italy, for helpful comments, and to the referees whose detailed comments improved the exposition.

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Correspondence to Steven J. Miller.

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Farris, M., Luntzlara, N., Miller, S.J. et al. Recurrence relations and Benford’s law. Stat Methods Appl 30, 797–817 (2021). https://doi.org/10.1007/s10260-020-00547-1

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### Keywords

• Benford’s law
• Recurrence relations

• 11K06
• 60F05
• 65Q30