In this article we show that Walsh-Hadamard transformations of generalized \(p\)-ary functions whose components are symmetric, rotation symmetric or a combination or concatenation of them are \(C\)-finite sequences. This result generalized many of the known results for regular \(p\)-ary functions. We also present a study of the roots of the characteristic polynomials related to these sequences and show that properties like balancedness and being bent are not shared by the underline \(p\)-ary functions.
