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

SIGMA 14 (2018), 077, 42 pages      arXiv:1801.07041      https://doi.org/10.3842/SIGMA.2018.077
Contribution to the Special Issue on Elliptic Hypergeometric Functions and Their Applications

### Connection Formula for the Jackson Integral of Type $A_n$ and Elliptic Lagrange Interpolation

Masahiko Ito a and Masatoshi Noumi b
a) Department of Mathematical Sciences, University of the Ryukyus, Okinawa 903-0213, Japan
b) Department of Mathematics, Kobe University, Rokko, Kobe 657-8501, Japan

Received January 23, 2018, in final form July 07, 2018; Published online July 24, 2018

Abstract
We investigate the connection problem for the Jackson integral of type $A_n$. Our connection formula implies a Slater type expansion of a bilateral multiple basic hypergeometric series as a linear combination of several specific multiple series. Introducing certain elliptic Lagrange interpolation functions, we determine the explicit form of the connection coefficients. We also use basic properties of the interpolation functions to establish an explicit determinant formula for a fundamental solution matrix of the associated system of $q$-difference equations.

Key words: Jackson integral of type $A_n$; $q$-difference equations; Selberg integral; Slater's transformation formulas; elliptic Lagrange interpolation.

pdf (662 kb)   tex (42 kb)

References

1. Aomoto K., On elliptic product formulas for Jackson integrals associated with reduced root systems, J. Algebraic Combin. 8 (1998), 115-126.
2. Aomoto K., Ito M., A determinant formula for a holonomic $q$-difference system associated with Jackson integrals of type $BC_n$, Adv. Math. 221 (2009), 1069-1114.
3. Aomoto K., Kato Y., A $q$-analogue of de Rham cohomology associated with Jackson integrals, in Special Functions (Okayama, 1990), ICM-90 Satell. Conf. Proc., Springer, Tokyo, 1991, 30-62.
4. Aomoto K., Kato Y., Connection formula of symmetric $A$-type Jackson integrals, Duke Math. J. 74 (1994), 129-143.
5. Askey R., Some basic hypergeometric extensions of integrals of Selberg and Andrews, SIAM J. Math. Anal. 11 (1980), 938-951.
6. Evans R.J., Multidimensional $q$-beta integrals, SIAM J. Math. Anal. 23 (1992), 758-765.
7. Forrester P.J., Log-gases and random matrices, London Mathematical Society Monographs Series, Vol. 34, Princeton University Press, Princeton, NJ, 2010.
8. Forrester P.J., Warnaar S.O., The importance of the Selberg integral, Bull. Amer. Math. Soc. (N.S.) 45 (2008), 489-534, arXiv:0710.3981.
9. Gasper G., Rahman M., Basic hypergeometric series, Encyclopedia of Mathematics and its Applications, Vol. 96, 2nd ed., Cambridge University Press, Cambridge, 2004.
10. Gustafson R.A., Multilateral summation theorems for ordinary and basic hypergeometric series in ${\rm U}(n)$, SIAM J. Math. Anal. 18 (1987), 1576-1596.
11. Habsieger L., Une $q$-intégrale de Selberg et Askey, SIAM J. Math. Anal. 19 (1988), 1475-1489.
12. Ishikawa M., Ito M., Okada S., A compound determinant identity for rectangular matrices and determinants of Schur functions, Adv. in Appl. Math. 51 (2013), 635-654, arXiv:1106.2915.
13. Ito M., Forrester P.J., A bilateral extension of the $q$-Selberg integral, Trans. Amer. Math. Soc. 369 (2017), 2843-2878, arXiv:1309.0001.
14. Ito M., Noumi M., A generalization of the Sears-Slater transformation and elliptic Lagrange interpolation of type $BC_n$, Adv. Math. 299 (2016), 361-380, arXiv:1506.07267.
15. Ito M., Sanada Y., On the Sears-Slater basic hypergeometric transformations, Ramanujan J. 17 (2008), 245-257.
16. Kadell K.W.J., A proof of Askey's conjectured $q$-analogue of Selberg's integral and a conjecture of Morris, SIAM J. Math. Anal. 19 (1988), 969-986.
17. Kadell K.W.J., A proof of some $q$-analogues of Selberg's integral for $k=1$, SIAM J. Math. Anal. 19 (1988), 944-968.
18. Kadell K.W.J., A simple proof of an Aomoto-type extension of Askey's last conjectured Selberg $q$-integral, J. Math. Anal. Appl. 261 (2001), 419-440.
19. Kaneko J., $q$-Selberg integrals and Macdonald polynomials, Ann. Sci. École Norm. Sup. (4) 29 (1996), 583-637.
20. Kaneko J., A $_1\Psi_1$ summation theorem for Macdonald polynomials, Ramanujan J. 2 (1998), 379-386.
21. Knop F., Sahi S., Difference equations and symmetric polynomials defined by their zeros, Int. Math. Res. Not. 1996 (1996), 473-486, math.QA/9610017.
22. Komori Y., Noumi M., Shiraishi J., Kernel functions for difference operators of Ruijsenaars type and their applications, SIGMA 5 (2009), 054, 40 pages, arXiv:0812.0279.
23. Macdonald I.G., Symmetric functions and Hall polynomials, 2nd ed., Oxford Mathematical Monographs, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1995.
24. Milne S.C., An elementary proof of the Macdonald identities for $A^{(1)}_l$, Adv. Math. 57 (1985), 34-70.
25. Milne S.C., A ${\rm U}(n)$ generalization of Ramanujan's $_1\Psi_1$ summation, J. Math. Anal. Appl. 118 (1986), 263-277.
26. Milne S.C., Schlosser M., A new $A_n$ extension of Ramanujan's ${}_1\psi_1$ summation with applications to multilateral $A_n$ series, Rocky Mountain J. Math. 32 (2002), 759-792, math.CA/0010162.
27. Mimachi K., Connection problem in holonomic $q$-difference system associated with a Jackson integral of Jordan-Pochhammer type, Nagoya Math. J. 116 (1989), 149-161.
28. Rosengren H., Elliptic hypergeometric series on root systems, Adv. Math. 181 (2004), 417-447, math.CA/0207046.
29. Selberg A., Remarks on a multiple integral, Norsk Mat. Tidsskr. 26 (1944), 71-78.
30. Slater L.J., General transformations of bilateral series, Quart. J. Math., Oxford Ser. (2) 3 (1952), 73-80.
31. Tannery J., Molk J., Éléments de la théorie des fonctions elliptiques. Tome III: Calcul intégral. Première partie, Gauthier-Villars, Paris, 1898.
32. Tarasov V., Varchenko A., Geometry of $q$-hypergeometric functions, quantum affine algebras and elliptic quantum groups, Astérisque 246 (1997), vi+135 pages, math.QA/9703044.
33. Warnaar S.O., $q$-Selberg integrals and Macdonald polynomials, Ramanujan J. 10 (2005), 237-268.
34. Warnaar S.O., Ramanujan's ${}_1\psi_1$ summation, Notices Amer. Math. Soc. 60 (2013), 18-22, arXiv:1206.2435.
35. Whittaker E.T., Watson G.N., A course of modern analysis, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1996.