Explicit formulae for the Hilbert symbol of a formal group over the Witt vectors

In this paper an explicit formula is obtained for a generalisation of the Hilbert symbol, associated with an arbitrary formal group of finite height, defined over the ring of Witt vectors with coefficients in a perfect field of characteristic $p>0$. This formula becomes the Bruckner–Vostokov formula in the case of a multiplicative formal group. The proof is based on an application of Fontaine's theory of $p$-adic periods of formal groups, the Fontaine–Wintenberg field-of-norms functor, and Witt's explicit reciprocity law in characteristic $p$.