Abstract
In this paper we shall provide a brief survey of the work begun by L. J. Rogers and W. N. Bailey which has led to an iterative method for producing infinite chains of (q-series identities. Apart from providing the reader with leads to the study of previous accomplishments, we shall emphasize the importance of examination of the seminal works in order to discern topics open to further development. This will lead us directly to a new construct: the Bailey tree.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
G. E. Andrews, On the q-analog of Rummer’s theorem and applications, Duke Math. J. 40 (1973), 525–528.
G. E. Andrews, An analytic generalization of the Rogers-Ramanujan identities for odd moduli, Proc. Nat. Acad. Sci. USA 71 (1974), 4082–4085.
G. E. Andrews, Problems and prospects for basic hypergeometric functions in: “Theory and Applications of Special Functions”, R. Askey, ed., Academic Press, New York, 1975, pp. 191–224.
G. E. Andrews, The Theory of Partitions, Encyl. of Math. and Its Appl., Vol. 2, Addison-Wesley, Reading, 1976; Reissued: Cambridge University Press, Cambridge, 1985 and 1998.
G. E. Andrews, Multiple series Rogers-Ramanujan type identities, Pacific J. Math. 114 (1984), 267–283.
G. E. Andrews, The fifth and seventh order mock theta functions, Trans. Amer. Math. Soc. 293 (1986), 113–134.
G. E. Andrews, q-Series: Their Development, CBMS Regional Conf. Lecture Series 66, Amer. Math. Soc, Providence, 1986.
G. E. Andrews, Umbral calculus, Bailey chains and pentagonal number theorems, J. Comb. Theory, Ser. A, 91 (2000), 464–475.
G. E. Andrews and A. Berkovich, A trinomial analogue of Bailey’s lemma and N = 2 superconformai invariance, Comm. in Math. Phys. 192 (1998), 245–260.
G. E. Andrews and D. Bowman, The Bailey transform and D. B. Sears, Quaest. Math. 22 (1999), 19–26.
G. E. Andrews, F. J. Dyson and D. Hickerson, Partitions and indefinite quadratic forms, Invent. Math. 91 (1988), 391–407.
G. E. Andrews, A. Schilling and S. O. Warnaar, An A 2 Bailey Lemma and Rogers-Ramanujan-type identities, J. Amer. Math. Soc. 12 (1999), 677–702.
W. N. Bailey, Some identities in combinatory analysis, Proc. London Math. Soc. (2) 49 (1947), 421–435.
W. N. Bailey, A transformation of nearly-poised basic hypergeometric series, J. London Math. Soc. 22 (1947), 237–240.
W. N. Bailey, Identities of the Rogers-Ramanujan type, Proc. London Math. Soc. (2) 50 (1949), 1–10.
A. Berkovich, B. M. McCoy and A. Schilling, N = 2 supersymmetry and Bailey pairs, Phys. A 228 (1996), 33–62.
J. M. and P. B. Borwein, Pi and the AGM, Wiley, New York, 1987.
D. M. Bressoud, Some identities for terminating q-series, Math. Proc. Camb. Phil. Soc. 81 (1981), 211–223.
D. M. Bressoud, An easy proof of the Rogers-Ramanujan identities, J. Number Theory 16 (1983), 235–241.
D. M. Bressoud, A matrix inverse, Proc. Amer. Math. Soc. 88 (1983), 446–448.
D. M. Bressoud, The Bailey lattice: an introduction, in: “Ramanujan Revisited”, G. E. Andrews et al, eds., Academic Press, New York 1988, pp. 57–67.
O. Foda and V.-H. Iuano, Virasoro character identities from the Andrews-Bailey construction, Int. J. Mod. Phys. A 12 (1997), 1651–1676.
G. Gasper and M. Rahman, Basic Hypergeometric Series, Encyl. of Math and Its Appl., Vol. 35, Cambridge University Press, Cambridge, 1990.
F. H. Jackson, Summation of q-hypergeometric series, Messenger of Math. 50 (1921), 101–112.
G. M. Lilly and S. C. Milne, The A l and C l Bailey transform and lemma, Bull. Amer. Math. Soc. (NS) 26 (1992), 258–263.
G. M. Lilly and S. C. Milne, The C l Bailey transform and Bailey lemma, Constr. Approx. 9 (1993), 473–500.
G. M. Lilly and S. C. Milne, Consequences of the A l and Ct Bai l ley transform and Bailey lemma, Discr. Math. 139 (1995), 319–346.
P. Paule, On identities of the Rogers-Ramanujan type, J. Math. Anal. and Appl. 107 (1985), 225–284.
P. Paule, The concept of Bailey chains, Sem. Lothar. Combin. B, 18f (1987), 24.
L. J. Rogers, Second memoir on the expansion of certain infinite products, Proc. London Math. Soc. 25 (1894), 318–343.
L. J. Rogers, On two theorems of combinatory analysis and allied identities, Proc. London Math. Soc. (2) 16 (1917), 315–336.
A. Schilling and S. O. Warnaar, A higher-level Bailey lemma, Int. J. Mod. Phys., B11 (1997), 189–195.
A. Schilling and S. O. Warnaar, A higher level Bailey lemma, proof and application, Ramanujan journal 2 (1998), 327–349.
A. Schilling and S. O. Warnaar, Conjugate Bailey pairs, to appear.
D. B. Sears, On the transformation theory of basic hypergeometric functions, Proc. London Math. Soc. (2) 53 (1951), 158–180.
L. J. Slater, A new proof of Rogers’ transformations of infinite series, Proc. London Math. Soc. (2) 53 (1951), 460–475.
L. J. Slater, Further identities of the Rogers-Ramanujan type, Proc. London Math. Soc. (2) 54 (1952), 147–167.
L. J. Slater, Generalized Hypergeometric Functions, Cambridge University Press, Cambridge, 1966.
O. Warnaar, A note on the trinomial analogue of Bailey’s lemma, J. Combin. Theory A 81 (1998), 114–118.
O. Warnaar, 50 years of Bailey’s lemma, Kerber Festschrift, to appear.
G. N. Watson, A new proof of the Rogers-Ramanujan identities, J. London Math. Soc., 4 (1929) pp. 4–9.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2001 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Andrews, G.E. (2001). Bailey’s Transform, Lemma, Chains and Tree. In: Bustoz, J., Ismail, M.E.H., Suslov, S.K. (eds) Special Functions 2000: Current Perspective and Future Directions. NATO Science Series, vol 30. Springer, Dordrecht. https://doi.org/10.1007/978-94-010-0818-1_1
Download citation
DOI: https://doi.org/10.1007/978-94-010-0818-1_1
Publisher Name: Springer, Dordrecht
Print ISBN: 978-0-7923-7120-5
Online ISBN: 978-94-010-0818-1
eBook Packages: Springer Book Archive