Six families of generalized hypergeometric series in a variable $x$ and an arbitrary number of parameters are considered. Each of them is indexed by an integer $n$. Linear recurrence relations in $n$ relate these functions and their product by the variable $x$. We give explicit factorizations of these equations as products of first order recurrence operators. Related recurrences are also derived for the derivative with respect to $x$. These formulas generalize well-known properties of the classical orthogonal polynomials.