June 2023 SOME SOLUTIONS TO VECTOR NONLINEAR RECURRENCE EQUATIONS
Rocky Mountain J. Math. 53(3): 969-981 (June 2023). DOI: 10.1216/rmj.2023.53.969

## Abstract

Withers and Nadarajah (2022, to appear) gave solutions to univariate nonlinear recurrence equations. We consider the vector case. Let $\mathcal{C}$ denote the complex numbers. Let $F(z):\mathcal{C}^q \to \mathcal{C}^q$ be any analytic function. Let $w \in \mathcal{C}^q$ be any fixed point of $\mathbfit{F}(\mathbfit{z})$, that is, $\mathbfit{F}(\mathbfit{w}) = \mathbfit{w}$. Set $\mathop \mathbfit{F}\limits^ (\mathbfit{z}) = d\mathbfit{F}(\mathbfit{z}) / dz^\prime \in \mathcal{C}^{q \times q}$. Then for any eigenvalue $r$ of $\mathop \mathbfit{F}\limits^ (\mathbfit{w})$, the recurrence equation

$z_{n + 1} = F(z_n ) \in \mathcal{C}^q ,$

for $n = 0,1,2, \ldots$, has a solution of the form

$\mathbfit{z}_n = \mathbfit{z}_n (\mathbfit{w},\alpha r^n ) = \mathbfit{w} + \mathop {\mathop \sum \nolimits^ }\limits_\infty ^{i = 1} \mathbfit{a}_i (\mathbfit{w})(\alpha r^n )^i ,$

where $\alpha \in \mathcal{C}$ is arbitrary and $\mathbfit{a}_i (\mathbfit{w}) \in \mathcal{C}^q$ are given by recurrence.

## Citation

Christopher S. Withers. Saralees Nadarajah. "SOME SOLUTIONS TO VECTOR NONLINEAR RECURRENCE EQUATIONS." Rocky Mountain J. Math. 53 (3) 969 - 981, June 2023. https://doi.org/10.1216/rmj.2023.53.969

## Information

Received: 30 May 2022; Revised: 11 July 2022; Accepted: 23 July 2022; Published: June 2023
First available in Project Euclid: 21 July 2023

MathSciNet: MR4617925
zbMATH: 07731159
Digital Object Identifier: 10.1216/rmj.2023.53.969

Subjects:
Primary: 65H20 , 65Q99

Keywords: Analytic function , Bell polynomial , recurrence relation