Addition theorems for $${\mathcal {C}}^k$$ C k real functions and applications in ordinary differential equations

This work establishes the existence of addition theorems and double-angle formulas for $${\mathcal {C}}^k$$ real scalar functions. Moreover, we determine necessary and sufficient conditions for a bivariate function to be an addition formula for a $${\mathcal {C}}^k$$ real function. The double-angle formulas allow us to generate a duplication algorithm, which can be used as an alternative to the classical numerical methods to obtain an approximation for the solution of an ordinary differential equation. We demonstrate that this algorithm converges uniformly in any compact domain contained in the maximal domain of that solution. Finally, we carry out some numerical simulations showing a good performance of the duplication algorithm when compared with standard numerical methods.