Differentiating a Linear Recursive Sequence

Consider a sequence of real-valued functions of a real variable given by a homogeneous linear recursion with differentiable coefficients. We show that if the functions in the sequence are differentiable, then the sequence of derivatives also satisfies a homogeneous linear recursion whose order is at most double the order of original recursion. Similarly to the well-known operations that determine the elementwise sum and product of two linear recursive sequences, the coefficient functions of our recursion for the derivatives are easily computable from the original coefficient functions and their derivatives by direct manipulation of the coefficients of the characteristic polynomial of the recursion, without determining the roots. A simple application, computing linear recursions for derivatives of orthogonal polynomials, is presented.