Transformations of the Hypergeometric 4F3 with One Unit Shift: A Group Theoretic Study
Abstract
:1. Introduction and Preliminaries
2. The Group Structure of the Unit Shift Transformations
3. The Subgroup of Generated by Known Transformations
4. Related Transformation
5. Summation Formulas
6. Proof of Lemma 1
Author Contributions
Funding
Conflicts of Interest
Appendix A
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References
- Andrews, G.E.; Askey, R.; Roy, R. Special Functions; Cambridge University Press: Cambridge, UK, 1999. [Google Scholar]
- Krattenthaler, C.; Srinivasa Rao, K. On group theoretical aspects, hypergeometric transformations and symmetries of angular momentum coefficients. In Symmetries in Science XI; Kluwer Acad. Publ.: Dordrecht, The Netherlands, 2005; pp. 355–375. [Google Scholar]
- Rao, K.S. Hypergeometric series and Quantum Theory of Angular Momentum. In Selected Topics in Special Functions; Agarwal, R.P., Manocha, H.L., Srinivasa Rao, K., Eds.; Allied Publishers Ltd.: New Delhi, India, 2001; pp. 93–134. [Google Scholar]
- Rao, K.S.; Doebner, H.D.; Natterman, P. Generalized hypergeometric series and the symmetries of 3 − j and 6 − j coefficients. In Number Theoretic Methods. Developments in Mathematics; Kanemitsu, S., Jia, C., Eds.; Springer: Boston, MA, USA, 2002; Volume 8, pp. 381–403. [Google Scholar]
- Rao, K.S.; Lakshminarayanan, V. Generalized Hypergeometric Functions, Transformations and Group Theoretical Aspects; IOP Science: Bristol, UK, 2018. [Google Scholar]
- Shpot, M.A.; Srivastava, H.M. The Clausenian hypergeometric function 3F2 with unit argument and negative integral parameter differences. Appl. Math. Comput. 2015, 259, 819–827. [Google Scholar]
- Formichella, M.; Green, R.M.; Stade, E. Coxeter group actions on 4F3(1) hypergeometric series. Ramanujan J. 2011, 24, 93–128. [Google Scholar] [CrossRef] [Green Version]
- Olver, F.W.J.; Lozier, D.W.; Boisvert, R.F.; Clark, C.W. (Eds.) NIST Handbook of Mathematical Functions; Cambridge University Press: Cambridge, UK, 2010. [Google Scholar]
- Beyer, W.A.; Louck, J.D.; Stein, P.R. Group theoretical basis of some identities for thegeneralized hypergeometric series. J. Math. Phys. 1987, 28, 497–508. [Google Scholar] [CrossRef]
- Hardy, G.H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work; AMS Chelsea Pub.: Providence, RI, USA, 1999; p. 111. [Google Scholar]
- Green, R.M.; Mishev, I.D.; Stade, E. Coxeter group actions and limits of hypergeometric series. Ramanujan J. 2020. [Google Scholar] [CrossRef]
- Mishev, I.D. Coxeter group actions on Saalschützian 4F3(1) series and very-well-poised 7F6(1) series. J. Math. Anal. Appl. 2012, 385, 1119–1133. [Google Scholar] [CrossRef] [Green Version]
- Rao, K.S.; Van der Jeugt, J.; Raynal, J.; Jagannathan, R.; Rajeswari, V. Group theoretical basis for the terminating 3F2(1) series. J. Phys. A Math. Gen. 1992, 25, 861–876. [Google Scholar] [CrossRef]
- Van der Jeugt, J.; Rao, K.S. Invariance groups of transformations of basic hypergeometric series. J. Math. Phys. 1999, 40, 6692–6700. [Google Scholar] [CrossRef]
- Nørlund, N.E. Hypergeometric functions. Acta Math. 1955, 94, 289–349. [Google Scholar] [CrossRef]
- Olsson, P.O.M. Analytic continuation of higher-order hypergeometric functions. J. Math. Phys. 1966, 7, 702–710. [Google Scholar] [CrossRef]
- Bühring, W. Generalized hypergeometric functions at unit argument. Proc. Am. Math. Soc. 1992, 114, 145–153. [Google Scholar] [CrossRef] [Green Version]
- Kim, Y.S.; Rathie, A.K.; Paris, R.B. On two Thomae-type transformations for hypergeometric series with integral parameter differences. Math. Commun. 2014, 19, 111–118. [Google Scholar]
- Karp, D.B.; Prilepkina, E.G. Beyond the beta integral method: transformation formulas for hypergeometric functions via Meijer’s G function. arXiv 2019, arXiv:1912.11266. [Google Scholar]
- Ebisu, A.; Iwasaki, K. Three-term relations for 3F2(1). J. Math. Anal. Appl. 2018, 463, 593–610. [Google Scholar] [CrossRef]
- Bailey, W.N. Generalized Hypergeometric Series; Stecherthafner Service Agency: New York, NY, USA; London, UK, 1964; Reprinted from: Cambridge Tracts in Mathematics and Mathematical Physics, 1935, Volume 32. [Google Scholar]
- Karp, D.B.; Prilepkina, E.G. Degenerate Miller-Paris transformations. Results Math. 2019, 74. [Google Scholar] [CrossRef] [Green Version]
- Çetinkaya, A.; Karp, D. Summation formulas for some hypergeometric and some digamma series. Commun. Korean Math. Soc. 2020. in preparation. [Google Scholar]
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Karp, D.; Prilepkina, E. Transformations of the Hypergeometric 4F3 with One Unit Shift: A Group Theoretic Study. Mathematics 2020, 8, 1966. https://doi.org/10.3390/math8111966
Karp D, Prilepkina E. Transformations of the Hypergeometric 4F3 with One Unit Shift: A Group Theoretic Study. Mathematics. 2020; 8(11):1966. https://doi.org/10.3390/math8111966
Chicago/Turabian StyleKarp, Dmitrii, and Elena Prilepkina. 2020. "Transformations of the Hypergeometric 4F3 with One Unit Shift: A Group Theoretic Study" Mathematics 8, no. 11: 1966. https://doi.org/10.3390/math8111966
APA StyleKarp, D., & Prilepkina, E. (2020). Transformations of the Hypergeometric 4F3 with One Unit Shift: A Group Theoretic Study. Mathematics, 8(11), 1966. https://doi.org/10.3390/math8111966