Abstract
We give an alternative proof of an elliptic summation formula of type \(BC_n\) by applying the fundamental \(BC_n\) invariants to the study of Jackson integrals associated with the summation formula.
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Notes
The values of \(W_{(1^r)}(x;q,p,t,a,b)\) and \(P^{*(1,n)}_{(1^r)}(x;a,b,q,t;p)\) at this point are explicitly given in [1, 5], while the value of \( E_r(a_1,a_2;z) \) is computed by
$$\begin{aligned} E_r(a_1,a_2;z_1,\ldots ,z_r,a_2t^{n-r-1},\ldots ,a_2)= \prod _{i=1}^r \frac{\theta (a_2t^{n-r}z_i;p)\theta (a_2t^{n-r}/z_i;p)}{\theta (a_1a_2t^{n-i};p)\theta (a_2t^{n-2r+i}/a_1;p)}. \end{aligned}$$
References
Coskun, H., Gustafson, R.A.: Well-poised Macdonald functions \(W_\lambda \) and Jackson coefficients \(\omega _\lambda \) on \(BC_n\). In: Jack, Hall–Littlewood and Macdonald polynomials, Contemporary Mathematics, vol. 417, pp.127–155. American Mathematical Society, Providence, RI (2006)
Frenkel, I.B., Turaev, V.G.: Elliptic solutions of the Yang–Baxter equation and modular hypergeometric functions. In: The Arnold–Gelfand Mathematical Seminars, pp. 171–204. Birkhäuser Boston, MA (1997)
Gasper, G., Rahman, M.: Basic Hypergeometric Series. Encyclopedia of Mathematics and its Applications, vol. 96, 2nd edn. Cambridge University Press, Cambridge (2004)
Ito, M.: A multiple generalization of Slater’s transformation formula for a very-well-poised-balanced \({}_{2r}\psi _{2r}\) series. Q. J. Math. 59, 221–235 (2008)
Rains, E.M.: \(BC_n\)-symmetric Abelian functions. Duke Math. J. 135, 99–180 (2006)
Rains, E.M.: Transformations of elliptic hypergeometric integrals. Ann. Math. 171, 169–243 (2010)
Rosengren, H.: A proof of a multivariable elliptic summation formula conjectured by Warnaar. In: \(q\)-Series with Applications to Combinatorics, Number Theory, and Physics (Urbana, IL, 2000), pp. 193–202. Contemporary Mathematics, vol. 291. American Mathematical Society, Providence, RI (2001)
Ruijsenaars, S.N.M.: First order analytic difference equations and integrable quantum systems. J. Math. Phys. 38, 1069–1146 (1997)
Schlosser, M.: Elliptic enumeration of nonintersecting lattice paths. J. Combin. Theory Ser. A 114, 505–521 (2007)
Spiridonov, V.P.: Theta hypergeometric integrals. Algebra Anal. 15, 161–215 (2003). (translation in St. Petersburg Math. J. 15(2004), 929–967)
Spiridonov, V.P., Warnaar, S.O.: New multiple \(_6\psi _6\) summation formulas and related conjectures. Ramanujan J. 25, 319–342 (2011)
van Diejen, J.F., Spiridonov, V.P.: An elliptic Macdonald–Morris conjecture and multiple modular hypergeometric sums. Math. Res. Lett. 7, 729–746 (2000)
Warnaar, S.O.: Summation and transformation formulas for elliptic hypergeometric series. Constr. Approx. 18, 479–502 (2002)
Acknowledgments
The authors would like to thank the anonymous referees who kindly provided valuable comments and suggestions for the improvement of our manuscript.
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Communicated by Erik Koelink.
This work was partially supported by JSPS Kakenhi Grants (S)24224001, (C)25400118, and (B)15H03626.
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Ito, M., Noumi, M. Derivation of a \(BC_n\) Elliptic Summation Formula via the Fundamental Invariants. Constr Approx 45, 33–46 (2017). https://doi.org/10.1007/s00365-016-9340-8
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DOI: https://doi.org/10.1007/s00365-016-9340-8
Keywords
- \(BC_n\) elliptic summation formula
- Jackson integrals
- Fundamental invariants
- Interpolation functions
- q-Difference equations