Integral representation of the n-th derivative in de Branges-Rovnyak spaces and the norm convergence of its reproducing kernel

In this paper, we give an integral representation for the boundary values of derivatives of functions of the de Branges--Rovnyak spaces $\HH(b)$, where $b$ is in the unit ball of $H^\infty(\CC_+)$. In particular, we generalize a result of Ahern--Clark obtained for functions of the model spaces $K_b$, where $b$ is an inner function. Using hypergeometric series, we obtain a nontrivial formula of combinatorics for sums of binomial coefficients. Then we apply this formula to show the norm convergence of reproducing kernel $k_{\omega,n}^b$ of the evaluation of $n$-th derivative of elements of $\HH(b)$ at the point $\omega$ as it tends radially to a point of the real axis.