  We derive a convergent expansion of the generalized hypergeometric function
${}_{p-1}F_p$ in terms of the Bessel functions ${}_{0}F_1$ that holds uniformly
with respect to the argument in any horizontal strip of the complex plane. We
further obtain a convergent expansion of the generalized hypergeometric
function ${}_{p}F_p$ in terms of the confluent hypergeometric functions
${}_{1}F_1$ that holds uniformly in any right half-plane. For both functions,
we make a further step and give convergent expansions in terms of
trigonometric, exponential and rational functions that hold uniformly in the
same domains. For all four expansions we present explicit error bounds. The
accuracy of the approximations is illustrated with some numerical experiments.
