In this paper we give an improvement of the degree of the homogeneous linear recurrence with integer coefficients
that exponential sums of symmetric Boolean functions satisfy. This improvement is tight. We also compute the
asymptotic behavior of symmetric Boolean functions and provide a formula that allows us to determine if a symmetric
Boolean function is asymptotically not balanced. In particular, when the degree of the symmetric function is
a power of two, then the exponential sum is much smaller than \(2^n\).
