  The research about Harmonic Analysis associated with Jacobi expansions
carried 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 normalized
Jacobi polynomials and $I$ is the identity operator, we define the
corresponding Littlewood-Paley-Stein $g_k^{(\alpha,\beta)}$-functions
associated with it and we prove an equivalence of norms with weights for them.
As a consequence, we deduce a result for Laplace type multipliers.

Classical Analysis and ODEs