Bulletin of the American Mathematical Society

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ISSN 1088-9485(e) ISSN 0273-0979(p)

The $A_\ell$ and $C_\ell$ Bailey transform and lemma

Author(s): Stephen C. Milne; Glenn M. Lilly
Journal: Bull. Amer. Math. Soc. 26 (1992), 258-263.
MSC (2000): Primary 33D70; Secondary 05A19, 11B65, 11P83
MathSciNet review: 1118702
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Abstract: We announce a higher-dimensional generalization of the Bailey Transform, Bailey Lemma, and iterative "Bailey chain" concept in the setting of basic hypergeometric series very well-poised on unitary ${A_\ell }$ or symplectic ${C_\ell }$ groups. The classical case, corresponding to ${A_1}$ or equivalently ${\text{U}}(2)$ , contains an immense amount of the theory and application of one-variable basic hypergeometric series, including elegant proofs of the Rogers-Ramanujan-Schur identities. In particular, our program extends much of the classical work of Rogers, Bailey, Slater, Andrews, and Bressoud.

References:

[1]: A. K. Agarwal, G. Andrews, and D. Bressoud, The Bailey lattice, J. Indian Math. Soc. 51 (1987), 57-73. MR 988309 (90i:11113)
[2]: G. E. Andrews, q-Series: Their development and application in analysis, number theory, combinatorics, physics and computer algebra, CBMS Regional Con. Ser. in Math., no. 66, Conf. Board Math. Sci., Washington, DC, 1986. MR 858826 (88b:11063)
[3]: -, Problems and prospects for basic hypergeometric functions, Theory and Applications of Special Functions (R. Askey, ed.), Academic Press, New York, 1975, pp. 191-224. MR 0399528 (53:3372)
[4]: -, Connection coefficient problems and partitions, Proc. Sympos. Pure Math., (D. Ray-Chaudhuri, ed.), vol. 34, Amer. Math. Soc., Providence, RI, 1979, pp. 1-24. MR 525316 (80c:33004)
[5]: -, Multiple series Rogers-Ramanujan type identities, Pacific J. Math. 114 (1984), 267-283. MR 757501 (86c:11084)
[6]: G. E. Andrews, R. J. Baxter, and P. J. Forrester, Eight-vertex SOS model and generalized Rogers-Ramanujan-type identities, J. Statist. Phys. 35 (1984), 193-266. MR 748075 (86a:82001)
[7]: G. E. Andrews, F. J. Dyson, and D. Hickerson, Partitions and indefinite quadratic forms, Invent. Math. 91 (1988), 391-407. MR 928489 (89f:11071)
[8]: W. N. Bailey, Identities of the Rogers-Ramanujan type, Proc. London Math. Soc. (2) 50 (1949), 1-10. MR 0025025 (9:585b)
[9]: R. J. Baxter, Exactly solved models in statistical mechanics, Academic Press, London and New York, 1982. MR 690578 (86i:82002a)
[10]: L. C. Biedenharn and J. D. Louck, Angular momentum in quantum physics: Theory and applications, Encyclopedia of Mathematics and Its Applications, (G.-C. Rota, ed.), vol. 8, Addison-Wesley, Reading, MA, 1981. MR 635121 (83a:81001)
[11]: -, The Racah-Winger algebra in quantum theory, Encyclopedia of Mathematics and Its Applications, (G.-C. Rota, ed.), vol. 9, Addison-Wesley, Reading, MA, 1981. MR 636504 (83d:81002)
[12]: G. Gasper and M. Rahman, Basic hypergeometric series, Encyclopedia of Mathematics and Its Applications, (G.-C. Rota, ed.), vol. 35, Cambridge University Press, Cambridge, 1990. MR 1052153 (91d:33034)
[13]: I. Gessel and D. Stanton, Applications of q-Lagrange inversion to basic hypergeometric series, Trans. Amer. Math. Soc. 277 (1983), 173-201. MR 690047 (84f:33009)
[14]: R. A. Gustafson, The Macdonald identities for affine root systems of classical type and hypergeometric series very well-poised on semi-simple Lie algebras, Ramanujan International Symposium on Analysis (December 26th to 28th, 1987, Pune, India) (N. K. Thakare, ed.), 1989, pp. 187-224. MR 1117471 (92k:33015)
[15]: W. J. Holman, III, Summation Theorems for hypergeometric series in ${\text{U}}(n)$ , SIAM J. Math. Anal. 11 (1980), 523-532. MR 572203 (81g:22028)
[16]: W. J. Holman III, L. C. Biedenharn, and J. D. Louck, On hypergeometric series well-poised in ${\text{SU(}}n{\text{)}}$ , SIAM J. Math. Anal. 7 (1976), 529-541. MR 0412504 (54:627)
[17]: G. M. Lilly and S. C. Milne, The ${C_\ell }$ Bailey transform and Bailey lemma, preprint.
[18]: S. C. Milne, An elementary proof of the Macdonald identities for $A_\ell ^{(1)}$ , Adv. in Math. 57 (1985), 34-70. MR 800859 (87e:17020)
[19]: -, Basic hypergeometric series very well-poised in ${\text{U(}}n{\text{)}}$ , J. Math. Anal. Appl. 122 (1987), 223-256. MR 874970 (88h:33007)
[20]: -, Balanced $_3{\phi _2}$ summation theorems for ${\text{U}}(n)$ basic hypergeometric series, (in preparation).
[21]: -,A ${\text{U(}}n{\text{)}}$ generalization of Bailey's lemma, in preparation.
[22]: P. Paule, Zwei neue Transformationen als elementare Anwendungen der q-Vandermonde Formel, Ph.D. thesis, 1982, University of Vienna.
[23]: L. J. Rogers, Second memoir on the expansion of certain infinite products, Proc. London Math. Soc. 25 (1894), 318-343.
[24]: -, On two theorems of combinatory analysis and some allied identities, Proc. London Math. Soc (2) 16 (1917), 315-336.
[25]: L. J. Slater, Generalized hypergeometric functions, Cambridge University Press, London and New York, 1966. MR 0201688 (34:1570)
[26]: F. J. W. Whipple, On well-poised series, generalized hypergeometric series having parameters in pairs, each pair with the same sum, Proc. London Math. Soc. (2) 24 (1924), 247-263.
[27]: -, Well-poised series and other generalized hypergeometric series, Proc. London Math. Soc. (2) 25 (1926), 525-544.

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