ABSTRACT We derive a convergent expansion of the generalized hypergeometric function in terms of the Bessel functions 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 in terms of the confluent hypergeometric functions that holds uniformly in any right half-plane. For both functions, we make a further step forward 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 by some numerical experiments.

中文翻译：

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广义超几何函数 p–1Fp 和 pFp 的一致收敛展开

摘要 我们根据贝塞尔函数推导出广义超几何函数的收敛展开，该函数对于复平面的任何水平条带中的自变量一致成立。我们进一步根据在任何右半平面中一致成立的汇合超几何函数获得广义超几何函数的收敛展开。对于这两个函数，我们又向前迈进了一步，并在三角函数、指数函数和有理函数方面给出了收敛扩展，这些函数在同一域中保持一致。对于所有四个扩展，我们都给出了明确的错误界限。一些数值实验说明了近似值的准确性。