  In this paper we prove weighted $\ell^p$-inequalities for variation and
oscillation operators defined by semigroups of operators associated with
discrete Jacobi operators. Also, we establish that certain maximal operators
involving sums of differences of discrete Jacobi semigroups are bounded on
weighted $\ell^p$-spaces. $\ell^p$-boundedness properties for the considered
operators provide information about the convergence of the semigroup of
operators defining them.
