Kernel method and system of functional equations

Introduced by Knuth and subsequently developed by Banderier et al., Prodinger, and others, the kernel method is a powerful tool for solving power series equations in the form of F (z, t) = A (z, t) F (z₀, t) + B (z, t) and several variations. Recently, Hou and Mansour [Q.-H. Hou, T. Mansour, Kernel Method and Linear Recurrence System, J. Comput. Appl. Math. (2007), (in press).] presented a systematic method to solve equation systems of two variables F (z, t) = A (z, t) F (z₀, t) + B (z, t), where A is a matrix, and F and B are vectors of rational functions in z and t. In this paper we generalize this method to another type of rational function matrices, i.e., systems of functional equations. Since the types of equation systems we are interested in arise frequently in various enumeration questions via generating functions, our tool is quite useful in solving enumeration problems. To illustrate this, we provide several applications, namely the recurrence relations with two indices, and counting descents in signed permutations.