  We carry on the investigation initiated in [15] : we describe new shuffle
products coming from some special functions and group them, along with other
products encountered in the literature, in a class of products, which we name
$\varphi$-shuffle products. Our paper is dedicated to a study of the latter
class, from a combinatorial standpoint. We consider first how to extend
Radford's theorem to the products in that class, then how to construct their
bi-algebras. As some conditions are necessary do carry that out, we study them
closely and simplify them so that they can be seen directly from the definition
of the product. We eventually test these conditions on the products mentioned
above.
