\lim_{z\to z_0} V_\mu (x,z)$ are the sources of the correlation functions of the boundary current operator $  J^\mu (x) \,\, (= \bar{q}\gamma^\mu q (x) \,)\,$. That means: $$\langle 0 \vert \, J_\mu (x) J_\nu (y) \,  \vert 0 \rangle \, =\, \frac{\delta}{\delta V^{0\mu}(x)} \frac{\delta}{\delta V^{0\nu}(y)}
 \exp  \left( - I_{on shell} \right)\,.$$

Working in momentum space in the coordinates $x^ \mu$, or equivalently taking a plane wave solution, the field $ V_\mu (p,z) $ can be decomposed for convenience into a source factor times a $z$ dependent factor $$V_\mu (p,z) \,=\, v (p,z) V^0_\mu ( p ) \,,
 \label{Bulktoboundary}$$ where $ v (p,z)   $ is usually called bulk to boundary propagator and satisfies the equation of motion: $$\partial_z \Big( \frac{ e^{-k^ 2 z^ 2 }} { z}  \partial_z v (p,z) \Big) + \frac{p^ 2 }{z} e^{-k^ 2 z^ 2 }  v (p,z) \,=\, 0   \,.
\label{BulktoboundaryEOM}$$ In order that the factor $ V^0_\mu ( p )$, defined in the decomposition of eq. (\[Bulktoboundary\]), works as the source of the correlators of gauge theory currents, calculated at $z = z_0 $, one must impose the boundary condition: $$v (p, z=z_0) = 1\,.
 \label{boundarycondition}$$ The solution of eq. (\[BulktoboundaryEOM\]) is a Tricomi function 