  We study Pad\'e interpolation problems on an additive grid, related to
additive difference ($d$-) Painlev\'e equations of type $E_7^{(1)}$,
$E_6^{(1)}$, $D_4^{(1)}$ and $A_3^{(1)}$. By choosing suitable Pad\'e problems,
we can derive time evolution equations, scalar Lax pairs of contiguous type and
determinant formulae of special solutions given in terms of hypergeometric
functions, for the corresponding $d$-Painlev\'e equations.
