The canonical transformations of the dynamical multiparameter systems as recurrence relations for the models on the grating

TL;DR

This paper develops a theory of recurrence relations for multi-parameter dynamical systems on gratings using canonical transformations, linking to zero curvature representations and exemplifying with hypergeometric functions.

Contribution

It introduces a new framework connecting canonical transformations with recurrence relations for multi-parameter systems on gratings, including zero curvature links.

Findings

Constructed recurrence relations for hypergeometric functions

Linked invariance conditions to grating knot parameters

Established connection with zero curvature representations

Abstract

The theory of recurrence relations of linear multi-component and multi-parameter systems on the basis of the canonical transformations theory of the dynamical systems' sets is constructed. The parameters of the grating's knots are defined from the condition of the invariance of the model under shifts along the grating. The connection with a zero curvature representation for models on the grating is installed. The examples of two- and three-parameter systems described by the hypergeometric functions M(\alpha,\beta,t) and M(\alpha,\beta,\xi,t) are considered in details. The canonical recurrence relations increasing and decreasing parameters {\alpha,\beta,\xi} for solutions of the corresponding equations are constructed.