Systolic online algorithms for the multiplication of uni-variate polynomials and of multiple precision integers are synthesised using a novel method based on the following functional (or inductive) view: a systolic array is a head processor followed by an identical tail array. The synthesis method consists in first unfolding the functional expression of the target function until the first four elements are separated, and then by projecting the remaining list expression into the scalar space in order to obtain the transition function of the individual processors. The method is implemented as a set of rewrite rules in the Theorema system, and it generates the description of the systolic arrays in a completely automatic manner, starting from the functional definitions of the arithmetic operations.