Appearance of Balancing and related number sequences in steady state probabilities of some Markov chains

TL;DR

This paper explores how balancing and Lucas-balancing numbers, solutions to a Diophantine equation, appear in the steady state probabilities of certain Markov chains, revealing new mathematical identities.

Contribution

It demonstrates the connection between balancing numbers and Markov chain steady states, providing novel identities involving the silver ratio.

Findings

Balancing numbers appear in Markov chain steady state probabilities.

A new identity relating balancing numbers and the silver ratio is derived.

The work links number theory with Markov chain analysis.

Abstract

Balancing and Lucas-balancing numbers are solutions of a Diophantine equation and satisfy a second order homogeneous recurrence relation. Interestingly, these numbers can be seen as numerators and denominators in the steady state probabilities of a class of transition probability matrices of Markov chains. An identity relating the balancing numbers and the silver ratio can be obtained as a byproduct.