Transcendental solution to linear coefficient non-homogeneous second order recurrence relation with constant non-homogenity

TL;DR

This paper derives an explicit transcendental solution for a specific non-homogeneous second order recurrence relation, analyzes its properties, and discusses implications for computational visualization.

Contribution

It provides an explicit transcendental solution for a class of non-homogeneous recurrence relations with constant negative non-homogeneity.

Findings

The minimal positive solution is explicitly calculated as a function of initial value a_1.

The sequence of solutions is transcendental when coefficients are rational.

The solution converges to zero at a rate of O(i^{-1}).

Abstract

Second order recurrence relations of real numbers arise form various applications in discrete time dynamical systems as well as in the context on Markov chains. Solutions to the recurrence relations are fully defined by the first two initial values as well as the recurrence formula. We calculate in this work explicitly as a function of $a_1$ the minimal positive solution $(a_i)_{i\in\mathbb{N}}$ to non-homogeneous second order recurrence relation with affine coefficients when the non-homogeneity is constant and negative, and the first initial value equals $a_0=0$. We show that rational coefficients lead to a sequence of transcendental numbers. Additionally, we prove that this sequence is the only bounded solution when varying $a_1$, converges to $0$ and obtain the convergence speed in $O(i^{-1})$. We comment in the last section further on the choice of rational parameters in the recurrence relation and we make a link to the impossibility of obtaining computer based visualizations of the minimal positive solution $(a_i)_{i\in\mathbb{N}}$.