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 Zh. Vychisl. Mat. Mat. Fiz., 2010, Volume 50, Number 3, Pages 407–422 (Mi zvmmf4839)

Generation of Kummer's second theorem with application

Yong Sup Kima, M. A. Rakhab, A. K. Rathiec

a Department of Mathematics Education, Wonkwang University, Iksan 570-749, Korea
b Mathematics Department, College of Science, Suez Canal University, Ismailia (41522) – Egypt
c Vedant College of Engineering and Technology, Village: TULSI, Post-Jakhmund, Dist. BUNDI-323021, Rajasthan State, India

Abstract: The aim of this research paper is to obtain single series expression of
$$e^{-x/2} _1F_1(\alpha; 2\alpha+i; x)$$
for $i=0$, $\pm1$, $\pm2$, $\pm3$, $\pm4$, $\pm5$, where $_1F_1(\cdot)$ is the function of Kummer. For $i=0$, we have the well known Kummer second theorem. The results are derived with the help of generalized Gauss second summation theorem obtained earlier by Lavoie et al. In addition to this, explicit expressions of
$$_2F_1[-2n, \alpha; 2\alpha+i; 2] and _2F_1[-2n-1, \alpha; 2\alpha+i; 2]$$
each for $i=0$, $\pm1$, $\pm2$, $\pm3$, $\pm4$, $\pm5$ are also given. For $i=0$, we get two interesting and known results recorded in the literature. As an applications of our results, explicit expressios of
$$e^{-x} _1F_1(\alpha; 2\alpha+i; x)\times _1F_1(\alpha; 2\alpha+j; x)$$
for $i$$j=0, \pm1, \pm2, \pm3, \pm4, \pm5 and$$ (1-x)^{-a} _2F_1(a, b; 2b+j; -\frac{2x}{1-x})$$for$j=0$,$\pm1$,$\pm2$,$\pm3$,$\pm4$,$\pm5$are given. For$i=j=0$and$j=0$, we respectively get the well known Preece identity and a well known quadratic transformation formula due to Kummer. The results derived in this paper are simple, interesting, easily established and may by useful in the applicable sciences. Key words: hypergeometric Gauss summation theorem, Dixon theorem, generalization of Kummer theorem, function of Kummer, generalized gipergeometric function. Full text: PDF file (245 kB) References: PDF file HTML file English version: Computational Mathematics and Mathematical Physics, 2010, 50:3, 387–402 Bibliographic databases: UDC: 519.65 Received: 27.11.2008 Revised: 02.12.2008 Language: Citation: Yong Sup Kim, M. A. Rakha, A. K. Rathie, “Generation of Kummer's second theorem with application”, Zh. Vychisl. Mat. Mat. Fiz., 50:3 (2010), 407–422; Comput. Math. Math. Phys., 50:3 (2010), 387–402 Citation in format AMSBIB \Bibitem{KimRakRat10} \by Yong~Sup~Kim, M.~A.~Rakha, A.~K.~Rathie \paper Generation of Kummer's second theorem with application \jour Zh. Vychisl. Mat. Mat. Fiz. \yr 2010 \vol 50 \issue 3 \pages 407--422 \mathnet{http://mi.mathnet.ru/zvmmf4839} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2681919} \transl \jour Comput. Math. Math. Phys. \yr 2010 \vol 50 \issue 3 \pages 387--402 \crossref{https://doi.org/10.1134/S0965542510030024} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-77951791118}  Linking options: • http://mi.mathnet.ru/eng/zvmmf4839 • http://mi.mathnet.ru/eng/zvmmf/v50/i3/p407  SHARE: Citing articles on Google Scholar: Russian citations, English citations Related articles on Google Scholar: Russian articles, English articles This publication is cited in the following articles: 1. Kim Y.S., Rathie A.K., “Applications of a generalized form of Gauss's second theorem to the series$F_3(2)$”, Math. Commun., 16:2 (2011), 481–489 2. Choi J., Rathie A.K., “Two formulas contiguous to a quadratic transformation due to Kummer with an application”, Hacet. J. Math. Stat., 40:6 (2011), 885–894 3. Kim Y.S., Choi J., Rathie A.K., “Two results for the terminating$_3F_2(2)$with applications”, Bull. Korean. Math. Soc., 49:3 (2012), 621–633 4. Kim Y.S., Rathie A.K., “Some Results for Terminating$ _2F_1(2)$Series”, J. Inequal. Appl., 2013, 365 5. Ali Sh., Lachhwani K., “On Some Reduction Formulas of Kampe de Feriet Function”, Proceeding of International Conference on Recent Trends in Applied Physics & Material Science (RAM 2013), AIP Conference Proceedings, 1536, eds. Bhardwaj S., Shekhawat M., Suthar B., Amer Inst Physics, 2013, 1343–1345 6. Rakha M.A., Awad M.M., Rathie A.K., “On an Extension of Kummer's Second Theorem”, Abstract Appl. Anal., 2013, 128458 7. Choi J., Rathie A.K., “Relations Between Lauricella's Triple Hypergeometric Function$ _AF_3(x, y, z)$and Exton's Function$X8\$”, Adv. Differ. Equ., 2013, 34
8. Ibrahim A.K., Rakha M.A., Rathie A.K., “On Certain Hypergeometric Identities Deducible by Using the Beta Integral Method”, Adv. Differ. Equ., 2013, 341
9. Shekhawat N., Rathie A.K., Prakash O., “On a quadratic transformation due to Kummer and its generalizations”, INTERNATIONAL CONFERENCE ON CONDENSED MATTER AND APPLIED PHYSICS (ICC 2015): Proceeding of International Conference on Condensed Matter and Applied Physics (Bikaner, India, 30–31 October 2015), AIP Conference Proceedings, 1728, eds. Shekhawat M., Bhardwaj S., Suthar B., Amer Inst Physics, 2016, 020683
10. Shani K., Choi J., Rathie A.K., “Aderivation of Two Quadratic Transformations Contiguous to That of Kummer Via a Differential Equation Approach”, Honam Math. J., 38:4 (2016), 693–699
11. Kim Y., Rathie A.K., Paris R.B., “Evaluations of Some Terminating Hypergeometric F-2(1)(2) Series With Applications”, Turk. J. Math., 42:5 (2018), 2563–2575
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