  In this paper we consider square functions (also called Littlewood-Paley
g-functions) associated to Hankel convolutions acting on functions in the
Bochner-Lebesgue space $L^p((0,\infty),B)$, where $B$ is a UMD Banach space. As
special cases we study square functions defined by fractional derivatives of
the Poisson semigroup for the Bessel operator
$\Delta_\lambda=-x^{-\lambda}\frac{d}{dx}x^{2\lambda}\frac{d}{dx}x^{-\lambda}$,
$\lambda >0$. We characterize the UMD property for a Banach space $B$ by using
$L^p((0,\infty),B)$-boundedness properties of g-functions defined by
Bessel-Poisson semigroups. As a by product we prove that the fact that the
imaginary power $\Delta_\lambda ^{iw}$, $w\in \mathbb{R}\setminus\{0\}$, of the
Bessel operator $\Delta_\lambda$ is bounded in $L^p ((0,\infty),B)$,
$1<p<\infty$, characterizes the UMD property for the Banach space $B$. As
applications of our results for square functions we establish the boundedness
in $L^p((0,\infty),B)$ of spectral multipliers $m(\Delta_\lambda)$ of Bessel
operators defined by functions $m$ which are holomorphic in sectors
$\Sigma_\vartheta$.
