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
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. ISBN9789380416137.
Rosengren, Hjalmar (2016). "Elliptic Hypergeometric Functions". arXiv:1608.06161 [math.CA].
References
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, ISBN978-0-8176-3883-2, MR 1429892
Gasper, George; Rahman, Mizan (2004), Basic hypergeometric series, Encyclopedia of Mathematics and its Applications, vol. 96 (2nd ed.), Cambridge University Press, ISBN978-0-521-83357-8, 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