The and Bailey transform and lemma
Authors:
Stephen C. Milne and Glenn M. Lilly
Journal:
Bull. Amer. Math. Soc. 26 (1992), 258263
MSC (2000):
Primary 33D70; Secondary 05A19, 11B65, 11P83
MathSciNet review:
1118702
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Abstract: We announce a higherdimensional generalization of the Bailey Transform, Bailey Lemma, and iterative "Bailey chain" concept in the setting of basic hypergeometric series very wellpoised on unitary or symplectic groups. The classical case, corresponding to or equivalently , contains an immense amount of the theory and application of onevariable basic hypergeometric series, including elegant proofs of the RogersRamanujanSchur identities. In particular, our program extends much of the classical work of Rogers, Bailey, Slater, Andrews, and Bressoud.
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Joan Slater, Generalized hypergeometric functions, Cambridge
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F. J. W. Whipple, On wellpoised series, generalized hypergeometric series having parameters in pairs, each pair with the same sum, Proc. London Math. Soc. (2) 24 (1924), 247263.
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, Wellpoised series and other generalized hypergeometric series, Proc. London Math. Soc. (2) 25 (1926), 525544.
 [1]
 A. K. Agarwal, G. Andrews, and D. Bressoud, The Bailey lattice, J. Indian Math. Soc. 51 (1987), 5773. MR 988309 (90i:11113)
 [2]
 G. E. Andrews, qSeries: 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. 191224. MR 0399528 (53:3372)
 [4]
 , Connection coefficient problems and partitions, Proc. Sympos. Pure Math., (D. RayChaudhuri, ed.), vol. 34, Amer. Math. Soc., Providence, RI, 1979, pp. 124. MR 525316 (80c:33004)
 [5]
 , Multiple series RogersRamanujan type identities, Pacific J. Math. 114 (1984), 267283. MR 757501 (86c:11084)
 [6]
 G. E. Andrews, R. J. Baxter, and P. J. Forrester, Eightvertex SOS model and generalized RogersRamanujantype identities, J. Statist. Phys. 35 (1984), 193266. MR 748075 (86a:82001)
 [7]
 G. E. Andrews, F. J. Dyson, and D. Hickerson, Partitions and indefinite quadratic forms, Invent. Math. 91 (1988), 391407. MR 928489 (89f:11071)
 [8]
 W. N. Bailey, Identities of the RogersRamanujan type, Proc. London Math. Soc. (2) 50 (1949), 110. 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, AddisonWesley, Reading, MA, 1981. MR 635121 (83a:81001)
 [11]
 , The RacahWinger algebra in quantum theory, Encyclopedia of Mathematics and Its Applications, (G.C. Rota, ed.), vol. 9, AddisonWesley, 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 qLagrange inversion to basic hypergeometric series, Trans. Amer. Math. Soc. 277 (1983), 173201. MR 690047 (84f:33009)
 [14]
 R. A. Gustafson, The Macdonald identities for affine root systems of classical type and hypergeometric series very wellpoised on semisimple Lie algebras, Ramanujan International Symposium on Analysis (December 26th to 28th, 1987, Pune, India) (N. K. Thakare, ed.), 1989, pp. 187224. MR 1117471 (92k:33015)
 [15]
 W. J. Holman, III, Summation Theorems for hypergeometric series in , SIAM J. Math. Anal. 11 (1980), 523532. MR 572203 (81g:22028)
 [16]
 W. J. Holman III, L. C. Biedenharn, and J. D. Louck, On hypergeometric series wellpoised in , SIAM J. Math. Anal. 7 (1976), 529541. MR 0412504 (54:627)
 [17]
 G. M. Lilly and S. C. Milne, The Bailey transform and Bailey lemma, preprint.
 [18]
 S. C. Milne, An elementary proof of the Macdonald identities for , Adv. in Math. 57 (1985), 3470. MR 800859 (87e:17020)
 [19]
 , Basic hypergeometric series very wellpoised in , J. Math. Anal. Appl. 122 (1987), 223256. MR 874970 (88h:33007)
 [20]
 , Balanced summation theorems for basic hypergeometric series, (in preparation).
 [21]
 ,A generalization of Bailey's lemma, in preparation.
 [22]
 P. Paule, Zwei neue Transformationen als elementare Anwendungen der qVandermonde 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), 318343.
 [24]
 , On two theorems of combinatory analysis and some allied identities, Proc. London Math. Soc (2) 16 (1917), 315336.
 [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 wellpoised series, generalized hypergeometric series having parameters in pairs, each pair with the same sum, Proc. London Math. Soc. (2) 24 (1924), 247263.
 [27]
 , Wellpoised series and other generalized hypergeometric series, Proc. London Math. Soc. (2) 25 (1926), 525544.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S027309791992002689
PII:
S 02730979(1992)002689
Article copyright:
© Copyright 1992
American Mathematical Society
