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In this paper, the authors investigate the case of discrete multipleorthogonal polynomials with two weights on the step line, which satisfy Pearsonequations. The discrete multiple orthogonal polynomials in question areexpressed in terms of tau functions, which are double Wronskians of generalizedhypergeometric series. The shifts in the spectral parameter for type II andtype I multiple orthogonal polynomials are described using banded matrices. Itis demonstrated that these polynomials offer solutions to multicomponentintegrable extensions of the nonlinear Toda equations. Additionally, the papercharacterizes extensions of the Nijhoff-Capel totally discrete Toda equations.The hypergeometric $\tau$-functions are shown to provide solutions to theseintegrable nonlinear equations. Furthermore, the authors explore Laguerre-Freudequations, nonlinear equations for the recursion coefficients, with aparticular focus on the multiple Charlier, generalized multiple Charlier,multiple Meixner II, and generalized multiple Meixner II cases.

Toda and Laguerre-Freud equations and tau functions for hypergeometric discrete multiple orthogonal polynomials