The and Bailey transform and lemma

Authors:
Stephen C. Milne and Glenn M. Lilly

Journal:
Bull. Amer. Math. Soc. **26** (1992), 258-263

MSC (2000):
Primary 33D70; Secondary 05A19, 11B65, 11P83

DOI:
https://doi.org/10.1090/S0273-0979-1992-00268-9

MathSciNet review:
1118702

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Abstract | References | Similar Articles | Additional Information

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 or symplectic groups. The classical case, corresponding to or equivalently , 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.

**[1]**A. K. Agarwal, G. E. Andrews, and D. M. Bressoud,*The Bailey lattice*, J. Indian Math. Soc. (N.S.)**51**(1987), 57–73 (1988). MR**988309****[2]**George E. Andrews,*𝑞-series: their development and application in analysis, number theory, combinatorics, physics, and computer algebra*, CBMS Regional Conference Series in Mathematics, vol. 66, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1986. MR**858826****[3]**George E. Andrews,*Problems and prospects for basic hypergeometric functions*, Theory and application of special functions (Proc. Advanced Sem., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1975) Academic Press, New York, 1975, pp. 191–224. Math. Res. Center, Univ. Wisconsin, Publ. No. 35. MR**0399528****[4]**George E. Andrews,*Connection coefficient problems and partitions*, Relations between combinatorics and other parts of mathematics (Proc. Sympos. Pure Math., Ohio State Univ., Columbus, Ohio, 1978) Proc. Sympos. Pure Math., XXXIV, Amer. Math. Soc., Providence, R.I., 1979, pp. 1–24. MR**525316****[5]**George E. Andrews,*Multiple series Rogers-Ramanujan type identities*, Pacific J. Math.**114**(1984), no. 2, 267–283. MR**757501****[6]**George E. Andrews, R. J. Baxter, and P. J. Forrester,*Eight-vertex SOS model and generalized Rogers-Ramanujan-type identities*, J. Statist. Phys.**35**(1984), no. 3-4, 193–266. MR**748075**, https://doi.org/10.1007/BF01014383**[7]**George E. Andrews, Freeman J. Dyson, and Dean Hickerson,*Partitions and indefinite quadratic forms*, Invent. Math.**91**(1988), no. 3, 391–407. MR**928489**, https://doi.org/10.1007/BF01388778**[8]**W. N. Bailey,*Identities of the Rogers-Ramanujan type*, Proc. London Math. Soc. (2)**50**(1948), 1–10. MR**0025025**, https://doi.org/10.1112/plms/s2-50.1.1**[9]**Rodney J. Baxter,*Exactly solved models in statistical mechanics*, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London, 1982. MR**690578****[10]**L. C. Biedenharn and J. D. Louck,*Angular momentum in quantum physics*, Encyclopedia of Mathematics and its Applications, vol. 8, Addison-Wesley Publishing Co., Reading, Mass., 1981. Theory and application; With a foreword by Peter A. Carruthers. MR**635121****[11]**Lawrence C. Biedenharn and James D. Louck,*The Racah-Wigner algebra in quantum theory*, Encyclopedia of Mathematics and its Applications, vol. 9, Addison-Wesley Publishing Co., Reading, Mass., 1981. With a foreword by Peter A. Carruthers; With an introduction by George W. Mackey. MR**636504****[12]**George Gasper and Mizan Rahman,*Basic hypergeometric series*, Encyclopedia of Mathematics and its Applications, vol. 35, Cambridge University Press, Cambridge, 1990. With a foreword by Richard Askey. MR**1052153****[13]**Ira Gessel and Dennis Stanton,*Applications of 𝑞-Lagrange inversion to basic hypergeometric series*, Trans. Amer. Math. Soc.**277**(1983), no. 1, 173–201. MR**690047**, https://doi.org/10.1090/S0002-9947-1983-0690047-7**[14]**R. A. Gustafson,*The Macdonald identities for affine root systems of classical type and hypergeometric series very-well-poised on semisimple Lie algebras*, Ramanujan International Symposium on Analysis (Pune, 1987) Macmillan of India, New Delhi, 1989, pp. 185–224. MR**1117471****[15]**Wayne J. Holman III,*Summation theorems for hypergeometric series in 𝑈(𝑛)*, SIAM J. Math. Anal.**11**(1980), no. 3, 523–532. MR**572203**, https://doi.org/10.1137/0511050**[16]**W. J. Holman III, L. C. Biedenharn, and J. D. Louck,*On hypergeometric series well-poised in 𝑆𝑈(𝑛)*, SIAM J. Math. Anal.**7**(1976), no. 4, 529–541. MR**0412504**, https://doi.org/10.1137/0507043**[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 𝐴⁽¹⁾_{𝑙}*, Adv. in Math.**57**(1985), no. 1, 34–70. MR**800859**, https://doi.org/10.1016/0001-8708(85)90105-7**[19]**S. C. Milne,*Basic hypergeometric series very well-poised in 𝑈(𝑛)*, J. Math. Anal. Appl.**122**(1987), no. 1, 223–256. MR**874970**, https://doi.org/10.1016/0022-247X(87)90355-6**[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 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]**Lucy Joan Slater,*Generalized hypergeometric functions*, Cambridge University Press, Cambridge, 1966. MR**0201688****[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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Additional Information

DOI:
https://doi.org/10.1090/S0273-0979-1992-00268-9

Article copyright:
© Copyright 1992
American Mathematical Society