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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 uniformlywith respect to the argument in any horizontal strip of the complex plane. Wefurther obtain a convergent expansion of the generalized hypergeometricfunction ${}_{p}F_p$ 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 oftrigonometric, exponential and rational functions that hold uniformly in thesame domains. For all four expansions we present explicit error bounds. Theaccuracy of the approximations is illustrated with some numerical experiments.

Uniformly convergent expansions for the generalized hypergeometric functions of the Bessel and Kummer types