work by Regge and Wheeler [@Regge:1957td] followed by Zerilli’s analysis [@Zerilli:1970se], the theory of linear perturbations is well defined. The metric perturbations are decomposed according to their transformation properties under two-dimensional rotations. They are classified as odd (axial or vector) perturbations and even (polar or scalar) ones. The two modes give rise to a master linear differential equation similar to the Schrödinger equation, from which, the stability can be derived, following the work by Vishveshwara [@Vishveshwara:1970cc], Price [@Price:1971fb] and Wald [@Wald:1979]. This formalism has also been extended to higher dimensions where an additional mode, dubbed tensor perturbations, appears [@Gibbons:2002pq]. The procedure to study these solutions is also well-known. Developed initially by [@Kodama:2003jz; @Ishibashi:2003ap], it has been applied to various solutions (see e.g. [@Gleiser:2005ra; @Takahashi:2010ye; @Gannouji:2013eka]).

Instead of studying a particular model, we intend to give the generic conditions of stability of static black holes in Horndeski model [@Horndeski:1974wa]. These models which appeared for the first time in 1974 didn’t attract much interest, until they were rediscovered in a different context: brane models. Extra dimensions provide an approach to modify gravity without abandoning the form of the action proposed in Einstein’s general relativity, such as Lovelock polynomial [@Lovelock:1971yv]. From a phenomenological 