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Uniformly convergent expansions for the generalized hypergeometric functions \(_{p-1}F_p\) and \(_pF_p\). (English) Zbl 1476.33004

The authors find approximations for the \((p-1,p)\) and \((p,p)\)-type generalized hypergeometric functions in terms of Bessel functions. At the end of the paper, the authors provide a discussion about the accuracy of these approximations.
A representative result of the paper is the following: For appropriate constants \(a\) and \(r\), and vectors, the approximation \[ p-1 F_{p}\left(\begin{array}{l} \mathbf{a} \\ \mathbf{b} \end{array} \mid-\frac{z^{2}}{4}\right)=\frac{\Gamma(\mathbf{b})}{\Gamma(\mathbf{a})} \sum_{n=0}^{N-1} g_{n}(\mathbf{a} ; \mathbf{b}) \frac{J_{\psi(\mathbf{a} ; \mathbf{b})+n-1}(z)}{(z / 2)^{\psi(\mathbf{a} ; \mathbf{b})+n-1}}+R_{N}(z). \] holds true. Here \(J_\nu\) denotes the Bessel function of the first kind, and the remainder term is bounded as follows: \[ \left|R_{N}(z)\right| \leq K \mathrm{e}^{|\gamma z|} \frac{\log ^{r-1}(N)}{N^{a+1 / 2}}. \]
A somewhat more complicated expression is also valid which contains only elementary functions.
For the \((p,p)\)-type hypergeometric function another approximation is provided which is similar to the above, but the Kummer functions need to be used. In this case an estimate in terms of elementary function is also presented.

MSC:

33C20 Generalized hypergeometric series, \({}_pF_q\)
41A58 Series expansions (e.g., Taylor, Lidstone series, but not Fourier series)
41A80 Remainders in approximation formulas

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References:

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