Zhurnal Vychislitel'noi Matematiki i Matematicheskoi Fiziki
General information
Latest issue
Impact factor

Search papers
Search references

Latest issue
Current issues
Archive issues
What is RSS

Zh. Vychisl. Mat. Mat. Fiz.:

Personal entry:
Save password
Forgotten password?

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

This article is cited in 11 scientific papers (total in 11 papers)

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) Ц 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

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
\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
\jour Comput. Math. Math. Phys.
\yr 2010
\vol 50
\issue 3
\pages 387--402

Linking options:
  • http://mi.mathnet.ru/eng/zvmmf4839
  • http://mi.mathnet.ru/eng/zvmmf/v50/i3/p407

    SHARE: VKontakte.ru FaceBook Twitter Mail.ru Livejournal Memori.ru

    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  mathscinet  zmath  isi
    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  mathscinet  zmath  isi
    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  crossref  mathscinet  zmath  isi  scopus
    4. Kim Y.S., Rathie A.K., “Some Results for Terminating $ _2F_1(2)$ Series”, J. Inequal. Appl., 2013, 365  crossref  mathscinet  isi  elib  scopus
    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  crossref  adsnasa  isi  scopus
    6. Rakha M.A., Awad M.M., Rathie A.K., “On an Extension of Kummer's Second Theorem”, Abstract Appl. Anal., 2013, 128458  crossref  mathscinet  zmath  isi  scopus
    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  crossref  mathscinet  zmath  isi  elib  scopus
    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  crossref  mathscinet  zmath  isi  scopus
    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  crossref  isi  scopus
    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  crossref  mathscinet  zmath  isi
    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  crossref  isi  scopus
  • ∆урнал вычислительной математики и математической физики Computational Mathematics and Mathematical Physics
    Number of views:
    This page:309
    Full text:76
    First page:6

    Contact us:
     Terms of Use  Registration to the website  Logotypes © Steklov Mathematical Institute RAS, 2021