# Algebraic independence of certain entire functions of two variables generated by linear recurrences

## TL;DR

This paper constructs a special entire function of two variables, generated by a linear recurrence, whose values and derivatives at algebraic points are algebraically independent, using Mahler functions theory.

## Contribution

It introduces a novel entire function with algebraic independence properties, linking linear recurrences and Mahler functions in multiple variables.

## Key findings

- Constructed an entire function with algebraic independence at algebraic points.
- Reduced algebraic independence to Mahler functions of several variables.
- Applied Mahler functions theory to prove the main result.

## Abstract

In this paper we construct an entire function of two variables having the property that its values and its partial derivatives of any order at any distinct algebraic points are algebraically independent. Such an entire function is generated by a linear recurrence. In order to prove this result, we reduce the algebraic independency to that of Mahler functions of several variables by shifting the linear recurrence and apply the theory of Mahler functions.