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 H(b), where b is in the unit ball of H(infinity) (C(+)). 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 evaluation of the n-th derivative of elements of R(b) at the point omega as it tends radially to a point of the real axis.