The genetic sums' representation for the moments of a system of Stieltjes functions and its application

This paper deals with the spectral problems for high-order nonsymmetric dif ference operators. The method of investigation is based on the analysis of the genetic sums formulas for the moments of the operator. The parameters o f these sums are shown to be connected with coefficients of the introduced vector Stieltjes continued fraction. The connections with vector orthogonal ity, Hermite-Pade approximation, and Hankel determinants are investigated. This gives a tool for the analysis of the solution of the direct and invers e spectral problem of the operator. It is applied to the integration of hie rarchy of the discrete KdV equations. The existence of a global solution is proved.