We showed that, under certain conditions, restricted and biased exponential sums and Walsh transforms of symmetric and rotation symmetric Boolean functions are, as in the case of non-biased domain, \(C\)-finite sequences. We also showed that under other conditions, these sequences are \(P\)-recursive, which is a somewhat different behavior than their non-biased counterparts. We also show that exponential sums and Walsh transforms of a family of rotation symmetric monomials over the restricted domain \(E_{n,j}\)=\(\{{\bf x}\in \mathbb{F}_2^n: wt({\bf x})=j\}\) (\(wt({\bf x})\) is the weight of the vector x) are given by polynomials of degree at most \(j\), and so, they are also \(C\)-finite sequences. Finally, we also present a study of the behavior of symmetric Boolean functions under these biased transforms.
