The author shows that the generalized hypergeometric function
satisfies a nonhomogeneous recurrence relation of order p, when is balanced, and of order otherwise. For
a homogeneous recurrence relation of order is given. The results are proved by using some general theorems due to J. Wimp [Math. Comput. 22, 363-373 (1968; Zbl 0186.104); ibid. 29, 577-581 (1975; Zbl 0304.33003)] and Y. Luke [The special functions and their approximations (1969; Zbl 0193.017)]. Some examples are given.