A Bailey lattice

Lovejoy, Jeremy

doi:10.1090/S0002-9939-03-07247-2

A Bailey lattice
HTML articles powered by AMS MathViewer

by Jeremy Lovejoy PDF

Proc. Amer. Math. Soc. 132 (2004), 1507-1516 Request permission

Abstract:

We exhibit a technique for generating new Bailey pairs which leads to deformations of classical $q$-series identities, multiple series identities of the Rogers-Ramanujan type, identities involving partial theta functions, and a variety of representations for $q$-series by number-theoretic objects such as weight 3/2 modular forms, ternary quadratic forms, and weighted binary quadratic forms.

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, 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
George E. Andrews, Ramanujan’s “lost” notebook. I. Partial $\theta$-functions, Adv. in Math. 41 (1981), no. 2, 137–172. MR 625891, DOI 10.1016/0001-8708(81)90013-X
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
George E. Andrews, Multiple series Rogers-Ramanujan type identities, Pacific J. Math. 114 (1984), no. 2, 267–283. MR 757501
George E. Andrews, Bailey chains and generalized Lambert series. I. Four identities of Ramanujan, Illinois J. Math. 36 (1992), no. 2, 251–274. MR 1156626
George E. Andrews, Freeman J. Dyson, and Dean Hickerson, Partitions and indefinite quadratic forms, Invent. Math. 91 (1988), no. 3, 391–407. MR 928489, DOI 10.1007/BF01388778
George E. Andrews and Dean Hickerson, Ramanujan’s “lost” notebook. VII. The sixth order mock theta functions, Adv. Math. 89 (1991), no. 1, 60–105. MR 1123099, DOI 10.1016/0001-8708(91)90083-J
D. M. Bressoud, The Bailey lattice: an introduction, Ramanujan revisited (Urbana-Champaign, Ill., 1987) Academic Press, Boston, MA, 1988, pp. 57–67. MR 938960
F. Franklin, Sur le développement du produit infini $(1-x)(1-x^2) (1-x^3)...$, Comptes Rendus 82 (1881), 448-450.
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
Jeremy Lovejoy, Lacunary partition functions, Math. Res. Lett. 9 (2002), no. 2-3, 191–198. MR 1909637, DOI 10.4310/MRL.2002.v9.n2.a5
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
C. J. Everett Jr., Annihilator ideals and representation iteration for abstract rings, Duke Math. J. 5 (1939), 623–627. MR 13
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, Partial theta functions I: beyond the lost notebook, preprint.