### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 12 (2016), 039, 21 pages      arXiv:1602.09027      https://doi.org/10.3842/SIGMA.2016.039
Contribution to the Special Issue on Orthogonal Polynomials, Special Functions and Applications

### Elliptic Hypergeometric Summations by Taylor Series Expansion and Interpolation

Michael J. Schlosser and Meesue Yoo
Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern-Platz 1, A-1090 Vienna, Austria

Received March 01, 2016, in final form April 13, 2016; Published online April 19, 2016

Abstract
We use elliptic Taylor series expansions and interpolation to deduce a number of summations for elliptic hypergeometric series. We extend to the well-poised elliptic case results that in the $q$-case have previously been obtained by Cooper and by Ismail and Stanton. We also provide identities involving S. Bhargava's cubic theta functions.

Key words: elliptic hypergeometric series; summations; Taylor series expansion; interpolation.

pdf (428 kb)   tex (25 kb)

References

1. Askey R., Wilson J., Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials, Mem. Amer. Math. Soc. 54 (1985), no. 319, iv+55 pages.
2. Baxter R.J., Eight-vertex model in lattice statistics and one-dimensional anisotropic Heisenberg chain. II. Equivalence to a generalized ice-type model, Ann. Physics 76 (1973), 25-47.
3. Bhargava S., Unification of the cubic analogues of the Jacobian theta-function, J. Math. Anal. Appl. 193 (1995), 543-558.
4. Bhargava S., Fathima S.N., Laurent coefficients for cubic theta functions, South East Asian J. Math. Math. Sci. 1 (2003), 27-31.
5. Cooper S., The Askey-Wilson operator and the ${}_6\Phi_5$ summation formula, South East Asian J. Math. Math. Sci. 1 (2002), 71-82.
6. Cooper S., Toh P.C., Determinant identities for theta functions, J. Math. Anal. Appl. 347 (2008), 1-7.
7. Date E., Jimbo M., Kuniba A., Miwa T., Okado M., Exactly solvable SOS models: local height probabilities and theta function identities, Nuclear Phys. B 290 (1987), 231-273.
8. Frenkel I.B., Turaev V.G., Elliptic solutions of the Yang-Baxter equation and modular hypergeometric functions, in The Arnold-Gelfand Mathematical Seminars, Editors V.I. Arnold, I.M. Gelfand, V.S. Retakh, M. Smirnov, Birkhäuser Boston, Boston, MA, 1997, 171-204.
9. Gasper G., Rahman M., Basic hypergeometric series, Encyclopedia of Mathematics and its Applications, Vol. 96, 2nd ed., Cambridge University Press, Cambridge, 2004.
10. Gessel I., Stanton D., Applications of $q$-Lagrange inversion to basic hypergeometric series, Trans. Amer. Math. Soc. 277 (1983), 173-201.
11. Ismail M.E.H., The Askey-Wilson operator and summation theorems, in Mathematical Analysis, Wavelets, and Signal Processing (Cairo, 1994), Contemp. Math., Vol. 190, Amer. Math. Soc., Providence, RI, 1995, 171-178.
12. Ismail M.E.H., Rains E.M., Stanton D., Orthogonality of very-well-poised series, Int. Math. Res. Not., to appear.
13. Ismail M.E.H., Stanton D., Applications of $q$-Taylor theorems, J. Comput. Appl. Math. 153 (2003), 259-272.
14. Ismail M.E.H., Stanton D., Some combinatorial and analytical identities, Ann. Comb. 16 (2012), 755-771.
15. Marco J.M., Parcet J., A new approach to the theory of classical hypergeometric polynomials, Trans. Amer. Math. Soc. 358 (2006), 183-214, math.CO/0312247.
16. Rains E.M., $BC_n$-symmetric Abelian functions, Duke Math. J. 135 (2006), 99-180, math.CO/0402113.
17. Rains E.M., Transformations of elliptic hypergeometric integrals, Ann. of Math. 171 (2010), 169-243, math.QA/0309252.
18. Rosengren H., Sklyanin invariant integration, Int. Math. Res. Not. 2004 (2004), 3207-3232, math.QA/0405072.
19. Rosengren H., An elementary approach to $6j$-symbols (classical, quantum, rational, trigonometric, and elliptic), Ramanujan J. 13 (2007), 131-166, math.CA/0312310.
20. Rosengren H., Schlosser M., Elliptic determinant evaluations and the Macdonald identities for affine root systems, Compos. Math. 142 (2006), 937-961, math.CA/0505213.
21. Schlosser M.J., A Taylor expansion theorem for an elliptic extension of the Askey-Wilson operator, in Special Functions and Orthogonal Polynomials, Contemp. Math., Vol. 471, Amer. Math. Soc., Providence, RI, 2008, 175-186, arXiv:0803.2329.
22. Schultz D., Cubic theta functions, Adv. Math. 248 (2013), 618-697.
23. Spiridonov V.P., Theta hypergeometric series, in Asymptotic Combinatorics with Application to Mathematical Physics (St. Petersburg, 2001), NATO Sci. Ser. II Math. Phys. Chem., Vol. 77, Editors V.A. Malyshev, A.M. Vershik, Kluwer Acad. Publ., Dordrecht, 2002, 307-327, math.CA/0303204.
24. Warnaar S.O., Summation and transformation formulas for elliptic hypergeometric series, Constr. Approx. 18 (2002), 479-502, math.QA/0001006.
25. Weber H., Elliptische Functionen und Algebraische Zahlen, Vieweg u. Sohn, Braunschweig, 1891.
26. Whittaker E.T., Watson G.N., A course of modern analysis, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1996.