It has been shown that a holomorphic function
$f$
in the unit ball
${{\mathbb{B}}_{n}}$
of
${{\mathbb{C}}_{n}}$
belongs to the weighted Bergman space
$A_{\alpha }^{p},\,p\,>\,n\,+\,1\,+\alpha $
, if and only if the function
$\left| f(z)\,-\,f(w) \right|/\left| 1\,-\,\left\langle z,\,w \right\rangle \right|$
is in
${{L}^{p}}({{\mathbb{B}}_{n}}\,\times \,{{\mathbb{B}}_{n}},\,d{{v}_{\beta }}\,\times \,d{{v}_{\beta }})$
, where
$\beta \,=\,(p\,+\,\alpha \,-\,n\,-\,1)/2$
and
$d{{v}_{\beta }}(z)\,=\,{{(1\,-\,{{\left| z \right|}^{2}})}^{\beta }}\,dv(z)$
. In this paper we consider the range
$0\,<\,p\,<\,n\,+\,1\,+\,\alpha $
and show that in this case,
$f\,\in \,A_{\alpha }^{p}\,(\text{i})$
(i) if and only if the function
$\left| f(z)\,-\,f(w) \right|/\left| 1\,-\,\left\langle z,\,w \right\rangle \right|$
is in
${{L}^{p}}({{\mathbb{B}}_{n}}\,\times \,{{\mathbb{B}}_{n}},\,d{{v}_{\alpha }}\,\times \,d{{v}_{\alpha }})$
, (ii) if and only if the function
$\left| f(z)\,-\,f(w) \right|/\left| z\,-\,w \right|$
is in
${{L}^{p}}({{\mathbb{B}}_{n}}\,\times \,{{\mathbb{B}}_{n}},\,d{{v}_{\alpha }}\,\times \,d{{v}_{\alpha }})$
. We think the revealed difference in the weights for the double integrals between the cases
$0\,<\,p\,<\,n\,+\,1\,+\,\alpha $
and
$p\,>\,n\,+\,1\,+\,\alpha $
is particularly interesting.