  A difference equation w.r.t. space-time dimension $d$ for $n$-point one-loop
integrals with arbitrary momenta and masses is introduced and a solution
presented. The result can in general be written as multiple hypergeometric
series with ratios of different Gram determinants as expansion variables.
Detailed considerations for $2-,3-$ and $4-$point functions are given. For the
$2-$ point function we reproduce a known result in terms of the Gauss
hypergeometric function $_2F_1$. For the $3-$point function an expression in
terms of $_2F_1$ and the Appell hypergeometric function $F_1$ is given. For the
$4-$point function a new representation in terms of $_2F_1$, $F_1$ and the
Lauricella-Saran functions $F_S$ is obtained. For arbitrary $d=4-2\epsilon$,
momenta and masses the $2-,3-$ and $4-$point functions admit a simple one-fold
integral representation. This representation will be useful for the calculation
of contributions from the $\epsilon-$ expansion needed in higher orders of
perturbation theory. Physically interesting examples of $3-$ and $4-$point
functions occurring in Bhabha scattering are investigated.
