We compute upper and lower bounds for the approximation of hyperbolic functions at points $1/s$$(s=1,2,\cdots ...$

We establish new combinatorial transcendence criteria for continued fraction expansions. Let $\alpha =[0;a_1,a_2,...]$ be an algebraic number of degree at least three. One of our criteria implies that the sequence of partial quotients ${\left(a_{\ell }\right)}_{\ell \ge 1}$ of $\alpha$ is not ‘too simple’ (in a suitable sense) and cannot be generated by a finite automaton.