Jacobi–Perron Algorithm and Recurrence Relations

This paper explores connections between multidimensional continued fractions, specifically the Jacobi–Perron algorithm, and generalized Fibonacci and Lucas sequences of arbitrary order r. Recurrence relations are established, a sufficient criterion for the convergence of the associated ratio vectors is proved, and the limit is identified explicitly in terms of the dominant root of the auxiliary equation. When the coefficient of the oldest term of the recurrence equals one, this limit is shown to be the initial vector of a purely periodic Jacobi–Perron algorithm. The results link the work of Bernstein, Williams, and related authors to fundamental recursive sequences.