In mathematics, an elliptic hypergeometric series is a series Σcn such that the ratio cn/cn−1 is an elliptic function of n, analogous to generalized hypergeometric series where the ratio is a rational function of n, and basic hypergeometric series where the ratio is a periodic function of the complex number n. They were introduced by Date-Jimbo-Kuniba-Miwa-Okado (1987) and Frenkel & Turaev (1997) in their study of elliptic 6-j symbols.
For surveys of elliptic hypergeometric series see Gasper & Rahman (2004), Spiridonov (2008) or Rosengren (2016).
Definitions[edit]
The q-Pochhammer symbol is defined by
The modified Jacobi theta function with argument x and nome p is defined by
The elliptic shifted factorial is defined by
The theta hypergeometric series r+1Er is defined by
The very well poised theta hypergeometric series r+1Vr is defined by
The bilateral theta hypergeometric series rGr is defined by
Definitions of additive elliptic hypergeometric series[edit]
The elliptic numbers are defined by
where the Jacobi theta function is defined by
The additive elliptic shifted factorials are defined by
The additive theta hypergeometric series r+1er is defined by
The additive very well poised theta hypergeometric series r+1vr is defined by
Further reading[edit]
- Spiridonov, V. P. (2013). "Aspects of elliptic hypergeometric functions". In Berndt, Bruce C. (ed.). The Legacy of Srinivasa Ramanujan Proceedings of an International Conference in Celebration of the 125th Anniversary of Ramanujan's Birth; University of Delhi, 17-22 December 2012. Ramanujan Mathematical Society Lecture Notes Series. Vol. 20. Ramanujan Mathematical Society. pp. 347–361. arXiv:1307.2876. Bibcode:2013arXiv1307.2876S. ISBN 9789380416137.
- Rosengren, Hjalmar (2016). "Elliptic Hypergeometric Functions". arXiv:1608.06161 [math.CA].
References[edit]
- Frenkel, Igor B.; Turaev, Vladimir G. (1997), "Elliptic solutions of the Yang-Baxter equation and modular hypergeometric functions", The Arnold-Gelfand mathematical seminars, Boston, MA: Birkhäuser Boston, pp. 171–204, ISBN 978-0-8176-3883-2, MR 1429892
- Gasper, George; Rahman, Mizan (2004), Basic hypergeometric series, Encyclopedia of Mathematics and its Applications, vol. 96 (2nd ed.), Cambridge University Press, ISBN 978-0-521-83357-8, MR 2128719
- Spiridonov, V. P. (2002), "Theta hypergeometric series", Asymptotic combinatorics with application to mathematical physics (St. Petersburg, 2001), NATO Sci. Ser. II Math. Phys. Chem., vol. 77, Dordrecht: Kluwer Acad. Publ., pp. 307–327, arXiv:math/0303204, Bibcode:2003math......3204S, MR 2000728
- Spiridonov, V. P. (2003), "Theta hypergeometric integrals", Rossiĭskaya Akademiya Nauk. Algebra i Analiz, 15 (6): 161–215, arXiv:math/0303205, Bibcode:2003math......3205S, doi:10.1090/S1061-0022-04-00839-8, MR 2044635, S2CID 14471695
- Spiridonov, V. P. (2008), "Essays on the theory of elliptic hypergeometric functions", Rossiĭskaya Akademiya Nauk. Moskovskoe Matematicheskoe Obshchestvo. Uspekhi Matematicheskikh Nauk, 63 (3): 3–72, arXiv:0805.3135, Bibcode:2008RuMaS..63..405S, doi:10.1070/RM2008v063n03ABEH004533, MR 2479997, S2CID 16996893
- Warnaar, S. Ole (2002), "Summation and transformation formulas for elliptic hypergeometric series", Constructive Approximation, 18 (4): 479–502, arXiv:math/0001006, doi:10.1007/s00365-002-0501-6, MR 1920282, S2CID 18102177