## Some solutions to vector nonlinear recurrence equations.(English)Zbl 07731159

Summary: Withers and Nadarajah (2022, to appear) gave solutions to univariate nonlinear recurrence equations. We consider the vector case. Let $$\mathscr{C}$$ denote the complex numbers. Let $$\boldsymbol{F}(\boldsymbol{z}):\mathscr{C}^q \to \mathscr{C}^q$$ be any analytic function. Let $$w \in \mathscr{C}^q$$ be any fixed point of $$\boldsymbol{F}(\boldsymbol{z})$$, that is, $$\boldsymbol{F}(\boldsymbol{w}) = \boldsymbol{w}$$. Set $$\dot{\boldsymbol{F}} (\boldsymbol{z}) = d\boldsymbol{F}(\boldsymbol{z}) / dz' \in \mathscr{C}^{q \times q}$$. Then for any eigenvalue $$r$$ of $$\dot{\boldsymbol{F}}(\boldsymbol{w})$$, the recurrence equation $z_{n + 1} = \boldsymbol{F}(z_n) \in \mathscr{C}^q,$ for $$n = 0, 1, 2, \ldots$$, has a solution of the form $\boldsymbol{z}_n = \boldsymbol{z}_n (\boldsymbol{w}, \alpha r^n) = \boldsymbol{w} + \sum^\infty_{i = 1} \boldsymbol{a}_i (\boldsymbol{w})(\alpha r^n)^i ,$ where $$\alpha \in \mathscr{C}$$ is arbitrary and $$\boldsymbol{a}_i (\boldsymbol{w}) \in \mathscr{C}^q$$ are given by recurrence.

### MSC:

 65H20 Global methods, including homotopy approaches to the numerical solution of nonlinear equations 65Q99 Numerical methods for difference and functional equations, recurrence relations

### Keywords:

analytic function; Bell polynomial; recurrence relation
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### References:

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