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The correspondence between dynamics of $\Delta$-Toda and $\Delta$-Volterra lattices for the coefficients of the Jacobi operator and its resolvent function is established. A method to solve inverse problem --integration of $\Delta$--Toda and $\Delta$-Volterra lattices -- based on Pad\'e approximants and continued fractions for the resolvent function is proposed. The main ingredient are orthogonal polynomials which satisfy an Appell condition, with respect to the forward difference operator $\Delta$. It is shown that the measure of orthogonality associated to these systems
of orthogonal polynomials evolve in $t$ like $(1+x^p)^{1-t}\mu(x)$, $p = 1,2$, where $\mu(x)$ is a given positive Borel measure. Moreover, the $\Delta$-Volterra lattice is related to the $\Delta$-Toda lattice from Miura or B\"{a}cklund transformations. Examples related with Jacobi and Laguerre orthogonal polynomials and $\Delta$-Toda equations are given, as well as explicit solutions of $\Delta$-Volterra and $\Delta$-Toda lattices.
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