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On an extension of Kummer’s second theorem. (English) Zbl 1277.33008

Summary: The aim of this paper is to establish an extension of Kummer’s second theorem in the form \(e^{-x/2}\,\,_0F_2 \left[ \begin{matrix} a, & 2+d;\\ 2a+2, & d;\end{matrix} x \right] =\, _0F_1 \left[ \begin{matrix} -;\\ a + 3/2; \end{matrix} x^2/16 \right] + ((a/d - 1/2)/(a + 1))x\,\,_0F_1 \left[ \begin{matrix} -;\\ a + 3/2;\end{matrix} x^2/16\right] + (cx^2/2(2a + 3))\,\,_0F_1 \left[ \begin{matrix} -;\\ a + 5/2; \end{matrix} x^2/16 \right]\), where \(c = (1/(a + 1))(1/2 - a/d) + a/d(d + 1)\), \(d \neq 0, -1, -2, \dots\) For \(d = 2a\), we recover Kummer’s second theorem. The result is derived with the help of Kummer’s second theorem and its contiguous results available in the literature. As an application, we obtain two general results for the terminating \(_3F_2(2)\) series. The results derived in this paper are simple, interesting, and easily established and may be useful in physics, engineering, and applied mathematics.

MSC:

33C20 Generalized hypergeometric series, \({}_pF_q\)
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[1] Kim, Y. S.; Rakha, M. A.; Rathie, A. K., Generalizations of Kummer’s second theorem with application, Journal of Computational Mathematics and Mathematical Physics, 50, 3, 387-402 (2010) · Zbl 1224.33001
[2] Kim, Y. S.; Rakha, M. A.; Rathie, A. K., Extensions of certain classical summation theorems for the series \(_2F_1,_3F_2\), and \(_4F_3\) with applications in Ramanujan’s summations, International Journal of Mathematics and Mathematical Sciences, 2010 (2010) · Zbl 1210.33012
[3] Rakha, M. A.; Rathie, A. K., Generalizations of classical summation theorems for the series \(_2F_1\) and \(_3F_2\) with applications, Integral Transforms and Special Functions, 22, 11, 823-840 (2011) · Zbl 1241.33006
[4] Bailey, W. N., Products of generalized hypergeometric series, Proceedings of the London Mathematical Society, 28, 1, 242-250 (1928) · JFM 54.0392.04
[5] Kummer, E. E., Über die hypergeometridche Reihe, Journal für Die Reine Und Angewandte Mathematik, 15, 39-83 (1836) · ERAM 015.0528cj
[6] Choi, J.; Rathie, A. K., Another proof of Kummer’s second theorem, Communications of the Korean Mathematical Society, 13, 4, 933-936 (1998) · Zbl 0968.33006
[7] Rainville, E. D., Special Functions (1960), New York, NY, USA: The Macmillan Company, New York, NY, USA · Zbl 0092.06503
[8] Rathie, A. K.; Nagar, V., On Kummer’s second theorem involving product of generalized hypergeometric series, Le Matematiche, 50, 1, 35-38 (1995) · Zbl 0842.33003
[9] Lavoie, J.-L.; Grondin, F.; Rathie, A. K., Generalizations of Watson’s theorem on the sum of a \(_3F_2\), Indian Journal of Mathematics, 34, 1, 23-32 (1992) · Zbl 0793.33005
[10] Rathie, A. K.; Pogány, T. K., New summation formula for \(_3F_2(1 / 2)\) and a Kummer-type II transformation of \(_2F_2(x)\), Mathematical Communications, 13, 1, 63-66 (2008) · Zbl 1146.33002
[11] Rakha, M. A., A note on Kummer-type II transformation for the generalized hypergeometric function \(_2F_2\), Mathematical Notes, 19, 1, 154-156 (2012) · Zbl 1286.33008
[12] Kim, Y. S.; Choi, J.; Rathie, A. K., Two results for the terminating \(_3F_2(2)\) with applications, Bulletin of the Korean Mathematical Society, 49, 3, 621-633 (2012) · Zbl 1258.33002
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