An elliptic $BC_n$ Bailey Lemma, multiple Rogers–Ramanujan identities and Euler’s Pentagonal Number Theorems
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Abstract:
An elliptic $BC_n$ generalization of the classical two parameter Bailey Lemma is proved, and a basic one parameter $BC_n$ Bailey Lemma is obtained as a limiting case. Several summation and transformation formulas associated with the root system $BC_n$ are proved as applications, including a $_6\varphi _5$ summation formula, a generalized Watson transformation and an unspecialized Rogers–Selberg identity. The last identity is specialized to give an infinite family of multilateral Rogers–Selberg identities. Standard determinant evaluations are then used to compute $B_n$ and $D_n$ generalizations of the Rogers–Ramanujan identities in terms of determinants of theta functions. Starting with the $BC_n$ $_6\varphi _5$ summation formula, a similar program is followed to prove an infinite family of $D_n$ Euler Pentagonal Number Theorems.References
- 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
- George E. Andrews, On the proofs of the Rogers-Ramanujan identities, $q$-series and partitions (Minneapolis, MN, 1988) IMA Vol. Math. Appl., vol. 18, Springer, New York, 1989, pp. 1–14. MR 1019838, DOI 10.1007/978-1-4684-0637-5_{1}
- George E. Andrews, Bailey’s transform, lemma, chains and tree, Special functions 2000: current perspective and future directions (Tempe, AZ), NATO Sci. Ser. II Math. Phys. Chem., vol. 30, Kluwer Acad. Publ., Dordrecht, 2001, pp. 1–22. MR 2006282, DOI 10.1007/978-94-010-0818-1_{1}
- George E. Andrews, Umbral calculus, Bailey chains, and pentagonal number theorems, J. Combin. Theory Ser. A 91 (2000), no. 1-2, 464–475. In memory of Gian-Carlo Rota. MR 1780034, DOI 10.1006/jcta.2000.3111
- George E. Andrews, The theory of partitions, Encyclopedia of Mathematics and its Applications, Vol. 2, Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1976. MR 0557013
- George Andrews and Alexander Berkovich, The WP-Bailey tree and its implications, J. London Math. Soc. (2) 66 (2002), no. 3, 529–549. MR 1934290, DOI 10.1112/S0024610702003617
- George E. Andrews, Anne Schilling, and S. Ole Warnaar, An $A_2$ Bailey lemma and Rogers-Ramanujan-type identities, J. Amer. Math. Soc. 12 (1999), no. 3, 677–702. MR 1669957, DOI 10.1090/S0894-0347-99-00297-0
- W. N. Bailey, Identities of the Rogers-Ramanujan type, Proc. London Math. Soc. (2) 50 (1948), 1–10. MR 25025, DOI 10.1112/plms/s2-50.1.1
- A. Berkovich and F. G. Garvan, Some observations on Dyson’s new symmetries of partitions, Journal of Combinatorial Theory, Series A, to appear, arXiv:math.CO/0203111.
- Alexander Berkovich and Peter Paule, Variants of the Andrews-Gordon identities, Ramanujan J. 5 (2001), no. 4, 391–404 (2002). MR 1891420, DOI 10.1023/A:1013995805667
- D. M. Bressoud, A matrix inverse, Proc. Amer. Math. Soc. 88 (1983), no. 3, 446–448. MR 699411, DOI 10.1090/S0002-9939-1983-0699411-9
- D. M. Bressoud, The Bailey lattice: an introduction, Ramanujan revisited (Urbana-Champaign, Ill., 1987) Academic Press, Boston, MA, 1988, pp. 57–67. MR 938960
- H. Coskun, A $BC_n$ Bailey lemma and generalizations of Rogers–Ramanujan identities, August 2003, Ph.D. thesis.
- Hasan Coskun and Robert A. Gustafson, Well-poised Macdonald functions $W_\lambda$ and Jackson coefficients $\omega _\lambda$ on $BC_n$, Jack, Hall-Littlewood and Macdonald polynomials, Contemp. Math., vol. 417, Amer. Math. Soc., Providence, RI, 2006, pp. 127–155. MR 2284125, DOI 10.1090/conm/417/07919
- H. Coskun, Elliptic and basic hypergeometric series summation and transformation identities associated to root systems, in preperation.
- H. Coskun, Andrews–Gordon Identities associated to root systems, in preparation.
- H. Coskun, Interpolation Bailey Lemma, in preparation.
- Freeman J. Dyson, A new symmetry of partitions, J. Combinatorial Theory 7 (1969), 56–61. MR 238711
- Euler, L. Evolutio producti infiniti $(1-x)(1-xx)(1-x^3)(1-x^4)(1-x^5)$ etc. in seriem simplicem., Acta Academiae Scientarum Imperialis Petropolitinae 1780, pp. 47-55, 1783.
- Igor B. Frenkel and Vladimir G. Turaev, Elliptic solutions of the Yang-Baxter equation and modular hypergeometric functions, The Arnold-Gelfand mathematical seminars, Birkhäuser Boston, Boston, MA, 1997, pp. 171–204. MR 1429892
- Kristina Garrett, Mourad E. H. Ismail, and Dennis Stanton, Variants of the Rogers-Ramanujan identities, Adv. in Appl. Math. 23 (1999), no. 3, 274–299. MR 1722235, DOI 10.1006/aama.1999.0658
- 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
- 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
- G. H. Hardy, Ramanujan. Twelve lectures on subjects suggested by his life and work, Cambridge University Press, Cambridge, England; The Macmillan Company, New York, 1940. MR 0004860
- James E. Humphreys, Introduction to Lie algebras and representation theory, Graduate Texts in Mathematics, Vol. 9, Springer-Verlag, New York-Berlin, 1972. MR 0323842
- C. Krattenthaler, Advanced determinant calculus, Sém. Lothar. Combin. 42 (1999), Art. B42q, 67. The Andrews Festschrift (Maratea, 1998). MR 1701596
- Stephen C. Milne and Glenn M. Lilly, The $A_l$ and $C_l$ Bailey transform and lemma, Bull. Amer. Math. Soc. (N.S.) 26 (1992), no. 2, 258–263. MR 1118702, DOI 10.1090/S0273-0979-1992-00268-9
- I. G. Macdonald, Affine root systems and Dedekind’s $\eta$-function, Invent. Math. 15 (1972), 91–143. MR 357528, DOI 10.1007/BF01418931
- I. G. Macdonald, The Poincaré series of a Coxeter group, Math. Ann. 199 (1972), 161–174. MR 322069, DOI 10.1007/BF01431421
- Stephen C. Milne, The $C_l$ Rogers-Selberg identity, SIAM J. Math. Anal. 25 (1994), no. 2, 571–595. MR 1266578, DOI 10.1137/S0036141092237480
- W. G. Morris, Constant term identities for finite and affine root systems: Conjectures and theorems, Ph.D. dissertation, Univ. of Wisconsin–Madison (1982).
- Andrei Okounkov, On Newton interpolation of symmetric functions: a characterization of interpolation Macdonald polynomials, Adv. in Appl. Math. 20 (1998), no. 4, 395–428. MR 1612846, DOI 10.1006/aama.1998.0590
- E. Rains, $BC_n$–symmetric abelian functions, math.CO$/$0402113.
- L. J. Rogers, Second memoir on the expansion of certain infinite products, Proc. London Math. Soc. 25 (1894), 318–343.
- Hjalmar Rosengren, Elliptic hypergeometric series on root systems, Adv. Math. 181 (2004), no. 2, 417–447. MR 2026866, DOI 10.1016/S0001-8708(03)00071-9
- Hjalmar Rosengren and Michael Schlosser, Summations and transformations for multiple basic and elliptic hypergeometric series by determinant evaluations, Indag. Math. (N.S.) 14 (2003), no. 3-4, 483–513. MR 2083087, DOI 10.1016/S0019-3577(03)90058-9
- Anne Schilling and S. Ole Warnaar, A higher level Bailey lemma: proof and application, Ramanujan J. 2 (1998), no. 3, 327–349. MR 1651423, DOI 10.1023/A:1009746932284
- Michael Schlosser, Summation theorems for multidimensional basic hypergeometric series by determinant evaluations, Discrete Math. 210 (2000), no. 1-3, 151–169. Formal power series and algebraic combinatorics (Minneapolis, MN, 1996). MR 1731612, DOI 10.1016/S0012-365X(99)00125-9
- L. J. Slater, Further identities of the Rogers-Ramanujan type, Proc. London Math. Soc. (2) 54 (1952), 147–167. MR 49225, DOI 10.1112/plms/s2-54.2.147
- John R. Stembridge, Hall-Littlewood functions, plane partitions, and the Rogers-Ramanujan identities, Trans. Amer. Math. Soc. 319 (1990), no. 2, 469–498. MR 986702, DOI 10.1090/S0002-9947-1990-0986702-5
- S. Ole Warnaar, 50 years of Bailey’s lemma, Algebraic combinatorics and applications (Gößweinstein, 1999) Springer, Berlin, 2001, pp. 333–347. MR 1851961
- S. O. Warnaar, Summation and transformation formulas for elliptic hypergeometric series, Constr. Approx. 18 (2002), no. 4, 479–502. MR 1920282, DOI 10.1007/s00365-002-0501-6
- G. N. Watson, A new proof of the Rogers–Ramanujan identities, J. London Math. Soc. 4 (1929), 4–9.
- V. P. Spiridonov, An elliptic incarnation of the Bailey chain, Int. Math. Res. Not. 37 (2002), 1945–1977. MR 1918235, DOI 10.1155/S1073792802205127
Additional Information
- Hasan Coskun
- Affiliation: Department of Mathematics, Binnion Hall, Room 314, Texas A&M University–Com- merce, Commerce, Texas 75429
- Email: hasan\_coskun@tamu-commerce.edu
- Received by editor(s): May 22, 2006
- Received by editor(s) in revised form: August 9, 2006, and October 16, 2006
- Published electronically: April 17, 2008
- © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 360 (2008), 5397-5433
- MSC (2000): Primary 05A19, 11B65; Secondary 05E20, 33D67
- DOI: https://doi.org/10.1090/S0002-9947-08-04457-7
- MathSciNet review: 2415079