Bäcklund transformations for discrete Painlevé equations: discrete P\(_{\text{II}}\)-P\(_{\text{V}}\)
The authors present an algorithm which is similar to the algorithm introduced by \textit{A. S. Fokas} and \textit{M. J. Ablowitz} [J. Math. Phys. 23, 2033--2042 (1982; Zbl 0504.34022)] for continuous Painlevé equations, to obtain the Bäcklund transformations for discrete \(\text{P}_{\text{II}}- \text{P}_{\text{V}}\). The algorithm is simple and based on the investigation of the discrete Riccati equation for the solution of a given discrete Painlevé equation, with coefficients depending linearly on the solution of another discrete equation of Painlevé type. The Miura transformation for d-P\(_{\text{II}}\) and d-P\(_{\text{IV}}\), and the Bäcklund transformation for d-P\(_{\text{II}}\) and d-P\(_{\text{IV}}\) that are presented in this work, were previously known. The special-function solutions for the discrete Painlevé equations are extensively covered in the literature. But the transformations for q-P\(_{\text{III}}\) and q-P\(_{\text{V}}\) were not discussed before. As an application of the algorithm, special solutions which are the discrete analogue of the classical special functions, for discrete P\(_{\text{II}}-\text{P}_{\text{V}}\) whenever the parameters satisfy certain conditions (the linearizability conditions) are presented.
scientific article; zbMATH DE number 5005579
scientific article