Series3 performs a series expansion around 0. Series3 is equivalent to Series, except that it applies Normal on the result and that some Series bugs are fixed.
Series3[f, e, n] is equivalent to Series3[f, {e, 0, n}].
Series3[(x (1 - x))^(\[Delta]/2), \[Delta], 1]\frac{1}{2} \delta \log ((1-x) x)+1
Series3[Gamma[x], x, 1] // FullSimplify\frac{1}{x}-\gamma +1