Symmetric Powers of Hyperbolic Forms and of Trace Forms on Symbol Algebras

Let $K$ be a field with characteristic different from 2 and let $S$ be a symbol algebra over $K$. We compute the symmetric powers of hyperbolic quadratic forms over $K$. Also, we compute the symmetric powers of the quadratic trace form of $S$. In both cases we apply a generalised form of the Vandermonde convolution in the course of the computations.