$\nu = \nu (k)$ and we introduced the shorthand notation $\otimes \equiv \int dy$. Here ${^{QQ}{\mit\Phi}}$ is known from Ref. [@Mue94], while ${^{QG}{\mit\Phi}}$ can be found with the help of the special conformal anomaly matrix in the quark-gluon channel. Fortunately, in this sector no subtle conformal symmetry breaking effects occur as it happens in the quark-quark transition amplitude. Thus, the special conformal anomaly can be obtained immediately and reads: $$\label{QGCAM}
{^{QG}\hat\gamma^{(c)}}
= - \frac{\alpha_s}{2 \pi} \hat{b}(l) \ {^{QG}\hat\gamma}.$$

Now we have all necessary results to solve the problem in question. First we outline the procedure for the evaluation of the coefficient function for the polarized non-forward gluon contribution and then we just list the results for other cases since the machinery we have developed is quite general and can be easily extended to the quark coefficient functions as well.

Polarized case.
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For the $QG$-sector we find from Eq. (\[QGCAM\]) that the correction to the eigenfunctions are completely expressed in terms of the shift operator: $$\label{Phi}
{^{QG}{\mit\Phi}} (x, y, \zeta)
= ({\cal I} - {\cal D}) S (x, z) \otimes {^{QG}\!K^A} ( z, y, \zeta ),$$ which generates the shift of the Gegenbauer polynomials index: $$S(x, y) \otimes [y \bar 