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The research about Harmonic Analysis associated with Jacobi expansionscarried out in \cite{ACL-JacI} and \cite{ACL-JacII} is continued in this paper.Given the operator $\mathcal{J}^{(\alpha,\beta)}=J^{(\alpha,\beta)}-I$, where$J^{(\alpha,\beta)}$ is the three-term recurrence relation for the normalizedJacobi polynomials and $I$ is the identity operator, we define thecorresponding Littlewood-Paley-Stein $g_k^{(\alpha,\beta)}$-functionsassociated with it and we prove an equivalence of norms with weights for them.As a consequence, we deduce a result for Laplace type multipliers.

Discrete Harmonic Analysis associated with Jacobi expansions III: the Littlewood-Paley-Stein $g_{k}$-functions and the Laplace type multipliers