Generalized exponential Vandermonde determinant and Hermite multipoint discrete boundary value problem

We introduce a determinant that encompasses the classical Vandermonde determinant, the generalized Vandermonde determinant, and the recently introduced exponential Vandermonde determinant when the exponents are nonnegative integers. An explicit factorization of such a determinant will be established. This factorization enables us to develop a computationally tractable necessary and sufficient condition for the existence of a unique solution of a Hermite (l-point) discrete boundary value problem.