Recurrences in Terms of Special Polynomials for Exponential Sums of Elementary Symmetric Polynomials over Finite Fields

It is known that exponential sums of symmetric Boolean functions are linear recurrent. The characteristic polynomial of the homogeneous linear recurrence that they satisfy can be expressed in terms of cyclotomic polynomials. In this work, we study the general recurrence for $q$-ary functions and, in the case of 3-ary functions, we express the characteristic polynomial of the recurrence in terms of some special polynomials