y]^{\nu - \frac{1}{2}}
C^\nu_j (2y - 1)
= \left. \frac{d}{d\rho} \right|_{\rho = 0}
[x \bar x]^{\nu - \frac{1}{2} + \rho} C^{\nu + \rho}_j (2x - 1).$$ In Eq. (\[Phi\]) ${\cal I}$ is an identity operator and ${\cal D}$ extracts the diagonal part in the expansion of any test function $\tau (x, y)$ with respect to $C^\nu_j$, i.e. $$\begin{aligned}
{\cal D}\tau (x, y) = \sum_{ j= 0 }^{\infty}
[x \bar x]^{ \nu - \frac{1}{2}}/ N_j (\nu) C^\nu_j(2x - 1)
\tau_{jj} C^\nu_j (2y - 1).\end{aligned}$$ The generalized evolution kernel[^5] $${^{QG}\!K^A} (x , x ', \zeta)
= 2 N_f T_F \Theta_{112}^1 ( x , x - \zeta, x - x ' )$$ can be diagonalized with the following identity $$\label{QGmixing}
\zeta
\int dy C_{j}^{\frac{3}{2}} \left( 2 \frac{y}{\zeta} - 1 \right)
{^{QG}\!K^A} ( y, x, \zeta )
= {^{QG}\gamma^A_j}
C_{j - 1}^{\frac{5}{2}}
\left( 2 \frac{x}{\zeta} - 1 \right) \ \mbox{with} \
{^{QG}\gamma^A_j} = - \frac{12 N_f T_F}{(j + 1)(j + 2)}.$$ The coefficient ${^{QG}\gamma^A_j}$ in front of the Gegenbauer polynomial coincides with the DGLAP moments of the kernel ${^{QG}\!K^A} ( y, x, 0 )$ up to the common factor $6/j$, which arises as a result of the conventional definition of the Gegenbauer polynomials, namely, 