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On an extension of Kummer-type II transformation. (English) Zbl 1311.33005

Summary: In the theory of hypergeometric and generalized hypergeometric series, Kummer’s type I und II transformations play an important role.
In this short research paper, we aim to establish the explicit expression of
\[ e^ {-\frac x2}\,_2F_2\left[\begin{matrix} a,&d+n;&\\ &&x\\ 2a+n,&d;&\end{matrix}\right] \]
for \(n=3\).
For \(n=0\), we have the well known Kummer’s second transformation. For \(n=1\), the result was established by A. K. Rathie and T. K. Pogany [Math. Commun. 13, No. 1, 63–66 (2008; Zbl 1146.33002)] and later on by A. K. Rathie and J. Choi [Commun. Korean Math. Soc. 13, No. 4, 933–936 (1998; Zbl 0968.33006)]. For \(n=2\), the result was recently established by M. A. Rakha et al. [J. Egypt. Math. Soc. 21, No. 3, 201–205 (2013; Zbl 1280.33004)]. The result is derived with the help of Kummer’s second transformation and its contiguous results recently obtained by Y. Sup Kim et al. [Zh. Vychisl. Mat. Mat. Fiz. 50, No. 1, 407–422 (2010) and Comput. Math., Math. Phys. 50, No. 3, 387–402 (2010; Zbl 1224.33001)]. The result established in this short research paper is simple, interesting, easily established and may be potentially useful.

MSC:

33C20 Generalized hypergeometric series, \({}_pF_q\)
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