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.

这项工作建立了\({\mathcal {C}}^k\)实标量函数的加法定理和双角公式的存在性。此外，我们确定了双变量函数是\({\mathcal {C}}^k\)实函数的加法公式的充分必要条件。双角公式允许我们生成重复算法，该算法可用作经典数值方法的替代方法，以获得常微分方程解的近似值。我们证明该算法在该解的最大域中包含的任何紧凑域中均匀收敛。最后，我们进行了一些数值模拟，表明与标准数值方法相比，复制算法具有良好的性能。