We establish a determinant formula for the bilinear form associated with the elliptic hypergeometric integrals of type BCn by studying the structure of q-difference equations to be satisfied by them. The determinant formula is proved by combining the q-difference equations of the determinant and its asymptotic analysis along the singularities. The elliptic interpolation functions of type BCn are essentially used in the study of the q-difference equations.
REFERENCES
1.
K.
Aomoto
, “q-analogue of de Rham cohomology associated with Jackson integrals, I
,” Proc. Jpn. Acad., Ser. A
66
, 161
–164
(1990
);K.
Aomoto
, “q-analogue of de Rham cohomology associated with Jackson integrals, II
,” Proc. Jpn. Acad., Ser. A
66
, 240
–244
(1990
).2.
K.
Aomoto
, “Finiteness of a cohomology associated with certain Jackson integrals
,” Tôhoku Math. J.
43
, 75
–101
(1991
).3.
K.
Aomoto
and M.
Ito
, “A determinant formula for a holonomic q-difference system associated with Jackson integrals of type BCn
,” Adv. Math.
221
, 1069
–1114
(2009
).4.
R.
Askey
and J.
Wilson
, “Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials
,” Mem. Am. Math. Soc.
54
(319
), iv+55
(1985
).5.
H.
Coskun
and R. A.
Gustafson
, “Well-poised Macdonald functions Wλ and Jackson coefficients ωλ on BCn
,” in Jack, Hall-Littlewood and Macdonald Polynomials
, Contemporary Mathematics Vol. 417 (American Mathematical Societys
, Providence, RI
, 2006
), pp. 127
–155
.6.
R. A.
Gustafson
, “Some q-beta integrals on SU(n) and Sp(n) that generalize the Askey-Wilson and Nasrallah-Rahman integrals
,” SIAM J. Math. Anal.
25
, 441
–449
(1994
).7.
R. A.
Gustafson
, “A generalization of Selberg’s beta integral
,” Bull. Am. Math. Soc.
22
, 97
–105
(1990
).8.
M.
Ishikawa
, M.
Ito
, and S.
Okada
, “A compound determinant identity for rectangular matrices and determinants of Schur functions
,” Adv. Appl. Math.
51
, 635
–654
(2013
).9.
M.
Ito
, “q-difference shift for a BCn-type Jackson integral arising from ‘elementary’ symmetric polynomials
,” Adv. Math.
204
, 619
–646
(2006
).10.
M.
Ito
, “Three-term relations between interpolation polynomials for a BCn-type basic hypergeometric series
,” Adv. Math.
226
, 4096
–4130
(2011
).11.
M.
Ito
and M.
Noumi
, “Derivation of a BCn elliptic summation formula via the fundamental invariants
, ” Constr. Approx.
45
, 33
–46
(2017
); e-print arXiv:1504.07018.12.
M.
Ito
and M.
Noumi
, “Evaluation of the BCn elliptic Selberg integral via the fundamental invariants
,” Proc. Am. Math. Soc.
145
, 689
–703
(2017
); e-print arXiv:1504.07317.13.
M.
Ito
and M.
Noumi
, “A generalization of the Sears–Slater transformation and elliptic Lagrange interpolation of type BCn
,” Adv. Math.
229
, 361
–380
(2016
); e-print arXiv:1506.07267.14.
Y.
Komori
, M.
Noumi
, and J.
Shiraishi
, “Kernel functions for difference operators of Ruijsenaars type and their applications
,” Symmetry Integrability Geom. Methods Appl.
5
, 054
(2009
).15.
B.
Nassrallah
and M.
Rahman
, “Projection formulas, a reproducing kernel and a generating function for q-Wilson polynomials
,” SIAM J. Math. Anal.
16
, 186
–197
(1985
).16.
A.
Okounkov
, “BC-type interpolation Macdonald polynomials and binomial formula for Koornwinder polynomials
,” Transform. Groups
3
, 181
–207
(1998
).17.
A.
Okounkov
, “Combinatorial formula for Macdonald polynomials and generic Macdonald polynomials, Transform
,” Groups
8
, 293
–305
(2003
).18.
E. M.
Rains
, “BCn-symmetric Abelian functions
,” Duke Math. J.
135
, 99
–180
(2006
).19.
E. M.
Rains
and V. P.
Spiridonov
, “Determinants of elliptic hypergeometric integrals
,” Funkt. Anal. Prilozhen.
43
, 67
–86
(2009
)E. M.
Rains
and V. P.
Spiridonov
[Funct. Anal. Appl.
43
, 297
–311
(2009
)].20.
V.
Tarasov
and A.
Varchenko
, “Geometry of q-hypergeometric functions, quantum affine algebras and elliptic quantum groups
,” Astérisque
246
, 139
(1997
).21.
J. F.
van Diejen
, “Self-dual Koornwinder–Macdonald polynomials
,” Invent. Math.
126
, 319
–339
(1996
).22.
J. F.
van Diejen
and V. P.
Spiridonov
, “Elliptic Selberg integrals
,” Int. Math. Res. Notices
2001
, 1083
–1110
.© 2019 Author(s).
2019
Author(s)
You do not currently have access to this content.
