(\theta) + k + 1 - z \big)^{-1} 
\sin \big( (j + 1) \theta \big) \sin \big( (\ell + 1) \theta \big) \, d\theta.
\end{split}$$ Now, implies that the action of the operator $A (-L + k + 1 - z)^{-1} B^\ast$ on $\ell^2(\mathcal{V})$ can be described by $$\label{eq4,13}
\begin{split}
& A (-L + k + 1 - z)^{-1} B^\ast \varphi (v) \\
& = \frac{1}{\pi} \sum_{n \, \geq \, 0} \sum_{\substack{j \, \geq \, 0 \\ \ell \, \geq \, 0}} 
\sum_{\substack{0 \leq m \leq N_j \\ 0 \leq q \leq N_\ell}} 
\biggl\langle \varphi,B E_m^{n,n+j} \biggr\rangle 
\Big( A E_q^{n,n+\ell} \Big) (v) 
\langle \chi_m^{n,n+j},\chi_q^{n,n+\ell} \rangle \\
& \times \int_{\mathbb{T}} \big( -2\sqrt{k} \cos (\theta) + k + 1 - z \big)^{-1} 
\sin \big( (j + 1) \theta \big) \sin \big( (\ell + 1) \theta \big) \, d\theta.
\end{split}$$ Since we have $$\biggl\langle \varphi,W E_m^{n,n+j} \biggr\rangle = 
\sum_{v' \in \mathcal{V}} \varphi (v') \overline{ \Big( W E_m^{n,n+j} \Big) (v')},$$ $W$ being as above, then it follows from that $$A (-L + k + 1 - z)^{-1} B^\ast 
\varphi (v) = \sum_{v' \in \mathcal{V}} K(v,v') \varphi (v'),$$ with $$\label{eq4,14}
\begin{split}
 K(v,v') :=  
& \frac{1}{\pi} \sum_{n \, \geq \, 0} \sum_{\substack{j \, \geq \, 0 