The relativistic oscillator $L$ is a relativistic version of the harmonic oscillator : in the Klein-Gordon symbolic calculus, it is straightforward to obtain the symbols of families of operators that commute with $L$. A Feynman integral type representation of $e^{-\varepsilon L}$, with especially nice properties, is derived as a first consequence ; also, in the one-dimensional case, one gets new exact properties or formulas relative to Mathieu functions.