  In this paper, we define a new product over $\mathbb{R}^{\infty}$, which
allows us to obtain a group isomorphic to $\mathbb R^*$ with the usual product.
This operation unexpectedly offers an interpretation of the R\'edei rational
functions, making more clear some of their properties, and leads to another
product, which generates a group structure over the Pell hyperbola. Finally, we
join together these results, in order to evaluate solutions of Pell equation in
an original way.
