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

## TL;DR

This paper develops uniformly convergent expansions for generalized hypergeometric functions of Bessel and Kummer types, providing explicit error bounds and numerical validation across complex domains.

## Contribution

It introduces new uniform convergent expansions for hypergeometric functions in terms of Bessel, confluent hypergeometric, trigonometric, exponential, and rational functions.

## Key findings

- Explicit error bounds are provided for all expansions.
- Numerical experiments demonstrate the accuracy of the approximations.
- The expansions hold uniformly in specified complex domains.

## Abstract

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.