SOME SOLUTIONS TO VECTOR NONLINEAR RECURRENCE EQUATIONS

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 $F(z)$, that is, $F(w)=w$. Set $\dot{F}(z)=dF(z)∕d{z}^{\prime }\in {\mathcal{𝒞}}^{q\times q}$. Then for any eigenvalue $r$ of $\dot{F}(w)$, the recurrence equation

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

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

$$z_n=z_n(w,\alpha r^n)=w+\sum _{i=1}^{\infty }{a}_i(w){(\alpha r^n)}^i,$$

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