1. Generalized discrete Lotka-Volterra equation, orthogonal polynomials and generalized epsilon algorithm.
- Author
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Chen, Xiao-Min, Chang, Xiang-Ke, He, Yi, and Hu, Xing-Biao
- Subjects
LOTKA-Volterra equations ,ORTHOGONAL polynomials ,DIVERGENT series ,LAX pair ,DISCRETE systems ,EQUATIONS of motion ,ALGORITHMS ,BACKLUND transformations - Abstract
In this paper, we propose a generalized discrete Lotka-Volterra equation and explore its connections with symmetric orthogonal polynomials, Hankel determinants and convergence acceleration algorithms. Firstly, we extend the fully discrete Lotka-Volterra equation to a generalized one with a sequence of given constants { u 0 (n) } and derive its solution in terms of Hankel determinants. Then, it is shown that the discrete equation of motion is transformed into a discrete Riccati system for a discrete Stieltjes function, hence leading to a complete linearization. Besides, we obtain its Lax pair in terms of symmetric orthogonal polynomials by generalizing the Christoffel transformation for the symmetric orthogonal polynomials. Moreover, a generalization of the famous Wynn's ε-algorithm is also derived via a Miura transformation to the generalized discrete Lotka-Volterra equation. Finally, the numerical effects of this generalized ε-algorithm are discussed by applying to some linearly, logarithmically convergent sequences and some divergent series. [ABSTRACT FROM AUTHOR]
- Published
- 2023
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