  Further results for conformal partial waves for four point functions for
conformal primary scalar fields in conformally invariant theories are obtained.
They are defined as eigenfunctions of the differential Casimir operators for
the conformal group acting on two variable functions subject to appropriate
boundary conditions. As well as the scale dimension $\Delta$ and spin $\ell$
the conformal partial waves depend on two parameters $a,b$ related to the
dimensions of the operators in the four point function. Expressions for the
Mellin transform of conformal partial waves are obtained in terms of
polynomials of the Mellin transform variables given in terms of finite sums.
Differential operators which change $a,b$ by $\pm 1$, shift the dimension $d$
by $\pm 2$ and also change $\Delta,\ell$ are found. Previous results for
$d=2,4,6$ are recovered. The trivial case of $d=1$ and also $d=3$ are also
discussed. For $d=3$ formulae for the conformal partial waves in some
restricted cases as a single variable integral representation based on the
Bateman transform are found.
