Elliptic well-poised Bailey transforms and lemmas on root systems. (English) Zbl 1387.33026
Summary: We list \(A_n\), \(C_n\) and \(D_n\) extensions of the elliptic WP Bailey transform and lemma, given for \(n=1\) by Andrews and Spiridonov. Our work requires multiple series extensions of Frenkel and Turaev’s terminating, balanced and very-well-poised \({}_{10}V_9\) elliptic hypergeometric summation formula due to Rosengren, and Rosengren and Schlosser. In our study, we discover two new \(A_n\) \({}_{12}V_{11}\) transformation formulas, that reduce to two new \(A_n\) extensions of Bailey’s \(_{10}\phi_9\) transformation formulas when the nome \(p\) is 0, and two multiple series extensions of Frenkel and Turaev’s sum.
MSC:
| 33D67 | Basic hypergeometric functions associated with root systems |
Keywords:
\(A_n\) elliptic and basic hypergeometric series; elliptic and basic hypergeometric series on root systems; well-poised Bailey transform and lemmaReferences:
| [1] | Agarwal, A. K. and Andrews, G. E. and Bressoud, D. M., The {B}ailey lattice, The Journal of the Indian Mathematical Society. New Series, 51, 57-73, (1987) · Zbl 0668.10020 |
| [2] | Andrews, George E. and Berkovich, Alexander, The {WP}-{B}ailey tree and its implications, Journal of the London Mathematical Society. Second Series, 66, 3, 529-549, (2002) · Zbl 1049.33010 · doi:10.1112/S0024610702003617 |
| [3] | Andrews, George E., Bailey’s transform, lemma, chains and tree, Special Functions 2000: Current Perspective and Future Directions ({T}empe, {AZ}), NATO Sci. Ser. II Math. Phys. Chem., 30, 1-22, (2001), Kluwer Acad. Publ., Dordrecht · Zbl 1005.33005 · doi:10.1007/978-94-010-0818-1_1 |
| [4] | Andrews, George E. and Schilling, Anne and Warnaar, S. Ole, An {\(A_2} {B\)}ailey lemma and {R}ogers–{R}amanujan-type identities, Journal of the American Mathematical Society, 12, 3, 677-702, (1999) · Zbl 0917.05010 · doi:10.1090/S0894-0347-99-00297-0 |
| [5] | Bailey, W. N., An identify involving {H}eine’s basic hypergeometric series, The Journal of the London Mathematical Society, 4, 4, 254-257, (1929) · JFM 55.0220.01 · doi:10.1112/jlms/s1-4.4.254 |
| [6] | Bailey, W. N., Some identities in combinatory analysis, Proceedings of the London Mathematical Society. Second Series, 49, 421-425, (1947) · Zbl 0041.03403 · doi:10.1112/plms/s2-49.6.421 |
| [7] | Bartlett, Nick and Warnaar, S. Ole, Hall–{L}ittlewood polynomials and characters of affine {L}ie algebras, Advances in Mathematics, 285, 1066-1105, (2015) · Zbl 1323.05130 · doi:10.1016/j.aim.2015.08.011 |
| [8] | Bhatnagar, Gaurav, Inverse relations, generalized bibasic series, and their {\({\rm U}(n)\)} extensions, (1995), The Ohio State University |
| [9] | Bhatnagar, Gaurav, {\(D_n\)} basic hypergeometric series, Ramanujan Journal. An International Journal Devoted to the Areas of Mathematics Influenced by Ramanujan, 3, 2, 175-203, (1999) · Zbl 0930.33012 · doi:10.1023/A:1006997424561 |
| [10] | Bhatnagar, G. and Schlosser, M., {\(C_n\)} and {\(D_n\)} very-well-poised {\({}_{10}\phi_9\)} transformations, Constructive Approximation. An International Journal for Approximations and Expansions, 14, 4, 531-567, (1998) · Zbl 0936.33009 · doi:10.1007/s003659900089 |
| [11] | Bressoud, D. M., A matrix inverse, Proceedings of the American Mathematical Society, 88, 3, 446-448, (1983) · Zbl 0529.05006 · doi:10.2307/2044991 |
| [12] | Br{\"u}nner, Frederic and Spiridonov, Vyacheslav P., A duality web of linear quivers, Physics Letters B, 761, 261-264, (2016) · Zbl 1368.81145 · doi:10.1016/j.physletb.2016.08.039 |
| [13] | Coskun, Hasan, An elliptic {\(BC_n} {B\)}ailey lemma, multiple {R}ogers–{R}amanujan identities and {E}uler’s pentagonal number theorems, Transactions of the American Mathematical Society, 360, 10, 5397-5433, (2008) · Zbl 1146.05006 · doi:10.1090/S0002-9947-08-04457-7 |
| [14] | Denis, R. Y. and Gustafson, R. A., An {\({\rm SU}(n)\)} {\(q\)}-beta integral transformation and multiple hypergeometric series identities, SIAM Journal on Mathematical Analysis, 23, 2, 552-561, (1992) · Zbl 0777.33009 · doi:10.1137/0523027 |
| [15] | van Diejen, Jan F., On certain multiple {B}ailey, {R}ogers and {D}ougall type summation formulas, Kyoto University. Research Institute for Mathematical Sciences. Publications, 33, 3, 483-508, (1997) · Zbl 0894.33007 · doi:10.2977/prims/1195145326 |
| [16] | Frenkel, Igor B. and Turaev, Vladimir G., Elliptic solutions of the {Y}ang–{B}axter equation and modular hypergeometric functions, The {A}rnold–{G}elfand Mathematical Seminars, 171-204, (1997), Birkh\"auser Boston, Boston, MA · Zbl 0974.17016 · doi:10.1007/978-1-4612-4122-5_9 |
| [17] | Gasper, George and Rahman, Mizan, Basic hypergeometric series, Encyclopedia of Mathematics and its Applications, 96, xx+287, (2004), Cambridge University Press, Cambridge · Zbl 0695.33001 · doi:10.1017/CBO9780511526251 |
| [18] | Griffin, Michael J. and Ono, Ken and Warnaar, S. Ole, A framework of {R}ogers–{R}amanujan identities and their arithmetic properties, Duke Mathematical Journal, 165, 8, 1475-1527, (2016) · Zbl 1405.11140 · doi:10.1215/00127094-3449994 |
| [19] | Gustafson, R. A., The {M}acdonald identities for affine root systems of classical type and hypergeometric series very-well-poised on semisimple {L}ie algebras, Ramanujan {I}nternational {S}ymposium on {A}nalysis ({P}une, 1987), 185-224, (1989), Macmillan of India, New Delhi |
| [20] | Jackson, F. H., Summation of {\(q\)}-hypergeometric series,, Messenger of Math., 50, 2, 101-112, (1920) · JFM 48.0415.01 |
| [21] | Jouhet, Fr\'ed\'eric, Shifted versions of the {B}ailey and well-poised {B}ailey lemmas, Ramanujan Journal. An International Journal Devoted to the Areas of Mathematics Influenced by Ramanujan, 23, 1-3, 315-333, (2010) · Zbl 1204.33021 · doi:10.1007/s11139-010-9282-x |
| [22] | Macdonald, I. G., Symmetric functions and {H}all polynomials, Oxford Classic Texts in the Physical Sciences, xii+475, (2015), The Clarendon Press, Oxford University Press, New York · Zbl 1332.05002 |
| [23] | McLaughlin, James and Sills, Andrew V. and Zimmer, Peter, Lifting {B}ailey pairs to {WP}-{B}ailey pairs, Discrete Mathematics, 309, 16, 5077-5091, (2009) · Zbl 1187.05016 · doi:10.1016/j.disc.2009.03.015 |
| [24] | Milne, S. C., An elementary proof of the {M}acdonald identities for {\(A^{(1)}_l\)}, Advances in Mathematics, 57, 1, 34-70, (1985) · Zbl 0586.33011 · doi:10.1016/0001-8708(85)90105-7 |
| [25] | Milne, S. C., Multiple {\(q\)}-series and {\({\rm U}(n)\)} generalizations of {R}amanujan’s {\(_1\Psi_1\)} sum, Ramanujan Revisited ({U}rbana-{C}hampaign, {I}ll., 1987), 473-524, (1988), Academic Press, Boston, MA · Zbl 0653.10012 |
| [26] | Milne, S. C., A {\(q\)}-analog of a {W}hipple’s transformation for hypergeometric series in {\({\rm U}(n)\)}, Advances in Mathematics, 108, 1, 1-76, (1994) · Zbl 0809.33010 · doi:10.1006/aima.1994.1065 |
| [27] | Milne, Stephen C., Balanced {\(_3\phi_2\)} summation theorems for {\({\rm U}(n)\)} basic hypergeometric series, Advances in Mathematics, 131, 1, 93-187, (1997) · Zbl 0886.33014 · doi:10.1006/aima.1997.1658 |
| [28] | Milne, Stephen C., A new {\({\rm U}(n)\)} generalization of the {J}acobi triple product identity, {\(q\)}-Series from a Contemporary Perspective ({S}outh {H}adley, {MA}, 1998), Contemp. Math., 254, 351-370, (2000), Amer. Math. Soc., Providence, RI · Zbl 0960.33009 · doi:10.1090/conm/254/03961 |
| [29] | Milne, Stephen C. and Lilly, Glenn M., Consequences of the {\(A_l\)} and {\(C_l} {B\)}ailey transform and {B}ailey lemma, Discrete Mathematics, 139, 1-3, 319-346, (1995) · Zbl 0870.33012 · doi:10.1016/0012-365X(94)00139-A |
| [30] | Milne, Stephen C. and Newcomb, John W., {\({\rm U}(n)\)} very-well-poised {\(_{10}\phi_9\)} transformations, Journal of Computational and Applied Mathematics, 68, 1-2, 239-285, (1996) · Zbl 0870.33013 · doi:10.1016/0377-0427(95)00248-0 |
| [31] | Rosengren, Hjalmar, Elliptic hypergeometric series on root systems, Advances in Mathematics, 181, 2, 417-447, (2004) · Zbl 1066.33017 · doi:10.1016/S0001-8708(03)00071-9 |
| [32] | Rosengren, Hjalmar, Elliptic hypergeometric functions · Zbl 1366.33013 |
| [33] | Rosengren, Hjalmar |
| [34] | Rosengren, Hjalmar, Gustafson–{R}akha-type elliptic hypergeometric series, SIGMA. Symmetry, Integrability and Geometry. Methods and Applications, 13, 037, 11 pages, (2017) · Zbl 1366.33013 · doi:10.3842/SIGMA.2017.037 |
| [35] | Rosengren, Hjalmar and Schlosser, Michael, Multidimensional matrix inversions and elliptic hypergeometric series on root systems · Zbl 1065.33014 |
| [36] | Schlosser, Michael, Multidimensional matrix inversions and {\(A_r\)} and {\(D_r\)} basic hypergeometric series, Ramanujan Journal. An International Journal Devoted to the Areas of Mathematics Influenced by Ramanujan, 1, 3, 243-274, (1997) · Zbl 0934.33006 · doi:10.1023/A:1009705129155 |
| [37] | Schlosser, Michael, Elliptic enumeration of nonintersecting lattice paths, Journal of Combinatorial Theory. Series A, 114, 3, 505-521, (2007) · Zbl 1119.05012 · doi:10.1016/j.jcta.2006.07.002 |
| [38] | Schlosser, Michael, A new multivariable {\(_6\psi_6\)} summation formula, Ramanujan Journal. An International Journal Devoted to the Areas of Mathematics Influenced by Ramanujan, 17, 3, 305-319, (2008) · Zbl 1183.33032 · doi:10.1007/s11139-007-9017-9 |
| [39] | Spiridonov, V. P., An elliptic incarnation of the {B}ailey chain, International Mathematics Research Notices, 2002, 37, 1945-1977, (2002) · Zbl 1185.33023 · doi:10.1155/S1073792802205127 |
| [40] | Spiridonov, V. P., Theta hypergeometric series, Asymptotic Combinatorics with Application to Mathematical Physics ({S}t. {P}etersburg, 2001), NATO Sci. Ser. II Math. Phys. Chem., 77, 307-327, (2002), Kluwer Acad. Publ., Dordrecht · Zbl 1041.11032 · doi:10.1007/978-94-010-0575-3_15 |
| [41] | Spiridonov, V. P., Bailey’s tree for integrals, Theoretical and Mathematical Physics, 139, 1, 104-111, (2004) · Zbl 1178.33019 · doi:10.1023/B:TAMP.0000022745.45082.18 |
| [42] | Spiridonov, V. P., Essays on the theory of elliptic hypergeometric functions, Russian Mathematical Surveys, 63, 3(381), 405-472, (2008) · Zbl 1173.33017 · doi:10.1070/RM2008v063n03ABEH004533 |
| [43] | Spiridonov, Vyacheslav P. and Warnaar, S. Ole, Inversions of integral operators and elliptic beta integrals on root systems, Advances in Mathematics, 207, 1, 91-132, (2006) · Zbl 1103.33015 · doi:10.1016/j.aim.2005.11.007 |
| [44] | Srivastava, H. M. and Singh, S. N. and Singh, S. P. and Yadav, Vijay, Certain derived {WP}-{B}ailey pairs and transformation formulas for {\(q\)}-hypergeometric series, Univerzitet u Ni\v su. Prirodno-Matemati\v cki Fakultet. Filomat, 31, 14, 4619-4628, (2017) · Zbl 1499.11030 |
| [45] | Warnaar, S. Ole, 50 years of {B}ailey’s lemma, Algebraic Combinatorics and Applications ({G}\"o\ss weinstein, 1999), 333-347, (2001), Springer, Berlin · Zbl 0972.11003 · doi:10.1007/978-3-642-59448-9_23 |
| [46] | Warnaar, S. Ole, Summation and transformation formulas for elliptic hypergeometric series, Constructive Approximation. An International Journal for Approximations and Expansions, 18, 4, 479-502, (2002) · Zbl 1040.33013 · doi:10.1007/s00365-002-0501-6 |
| [47] | Warnaar, S. Ole, Extensions of the well-poised and elliptic well-poised {B}ailey lemma, Koninklijke Nederlandse Akademie van Wetenschappen. Indagationes Mathematicae. New Series, 14, 3-4, 571-588, (2003) · Zbl 1054.33013 · doi:10.1016/S0019-3577(03)90061-9 |
| [48] | Warnaar, S. Ole |
| [49] | Watson, G. N., A new proof of the {R}ogers–{R}amanujan identities, The Journal of the London Mathematical Society, 4, 1, 4-9, (1929) · JFM 55.0219.09 · doi:10.1112/jlms/s1-4.1.4 |
| [50] | Whittaker, E. T. and Watson, G. N., A course of modern analysis, Cambridge Mathematical Library, vi+608, (1996), Cambridge University Press, Cambridge · Zbl 0951.30002 · doi:10.1017/CBO9780511608759 |
| [51] | Zhang, Z. and Huang, J., The \({C}_n-{B\) · Zbl 1438.33032 |
| [52] | Zhang, Zhizheng and Liu, Qiyan, {\( \text{U}(n+1)\)} {WP}-{B}ailey tree, Ramanujan Journal. An International Journal Devoted to the Areas of Mathematics Influenced by Ramanujan, 40, 3, 447-462, (2016) · Zbl 1350.33033 · doi:10.1007/s11139-016-9790-4 |
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