Abstract
By recursive use of the q-Saalschütz summation formula, we investigate further the Saalschütz chain reactions introduced by the author in (Chu, 2002). Some general series transformations which express basic terminating series in terms of finite multiple sums will be established. As applications, we derive by means of Jackson's 6ϕ5-series identity three transformations including one due to Andrews (1975). These transformations yield further a number of multiple Rogers-Ramanujan identities, whose research was initiated and developed mainly by Andrews and Bressoud from the middle of seventieth up to now.
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
Agarwal, A. K., Andrews, G. E., and Bressoud, D. M. (1987). The Bailey lattice. J. Indian Math. Soc. (N.S.), 51:57–73.
Agarwal, A. K. and Bressoud, D. M. (1989). Lattice paths and multiple basic hyper-geometric series. Pacific J. Math., 136:209–228.
Andrews, G. E. (1974). An analytic generalization of the Rogers-Ramanujan identities for odd moduli. Proc. Nat. Acad. Sci. U.S.A., 71:4082–4085.
Andrews, G. E. (1975). Problems and prospects for basic hypergeometric functions. In Askey, R., editor, The Theory and Applications of Special Functions, pages 191–224. Academic Press, New York.
Andrews, G. E. (1976). The Theory of Partitions, volume 2 of Encyclopedia of Math. and Appl. Cambridge University Press, Cambridge.
Andrews, G. E. (1979a). Connection coefficient problems and partitions. In Relations between combinatorics and other parts of mathematics (Proc. Sympos. Pure Math., Ohio State Univ., Columbus, Ohio, 1978), Proc. Sympos. Pure Math., XXXIV, pages 1–24. Amer. Math. Soc., Providence, RI.
Andrews, G. E. (1979b). Partitions and Durfee dissection. Amer. J. Math., 101(3):735–742.
Andrews, G. E. (1981). Multiple q-series identities. Houston J. Math., 7(1):11–22.
Andrews, G. E. (1984). Multiple series Rogers-Ramanujan type identities. Pacific J. Math., 114(2):267–283.
Andrews, G. E. (1986). q-Series: Their Development and Application in Analysis, Number Theory, Combinatorics, Physics, and Computer Algebra, volume 66 of CBMS Regional Conference Series in Mathematics. Published for the Conference Board of the Mathematical Sciences, Washington, DC.
Bailey, W. N. (1935). Generalized Hypergeometric Series. Cambridge University Press, Cambridge.
Bailey, W. N. (1947). Some identities in combinatory analysis. Proc. London Math. Soc. (2), 49:421–435.
Bailey, W. N. (1948). Identities of the Rogers-Ramanujan type. Proc. London Math. Soc. (2), 50:1–10.
Bressoud, D. M. (1980a). Analytic and combinatorial generalizations of the Rogers-Ramanujan identities. Mem. Amer. Math. Soc., 24(227):54.
Bressoud, D. M. (1980b). An analytic generalization of the Rogers-Ramanujan identities with interpretation. Quart. J. Math. Oxford Ser. 2, 31:385–399.
Bressoud, D. M. (1981). On partitions, orthogonal polynomials and the expansion of certain infinite products. Proc. London Math. Soc. (3), 42:478–500.
Bressoud, D. M. (1988). The Bailey lattice: an introduction. In Ramanujan Revisited (Urbana-Champaign, Ill., 1987), pages 57–67. Academic Press, Boston, MA.
Bressoud, D. M. (1989). Lattice paths and the Rogers-Ramanujan identities. In Number theory, Madras 1987, volume 1395 of Lecture Notes in Mathematics, pages 140–172. Springer-Verlag, Berlin-New York.
Bressoud, D. M., Ismail, M., and Stanton, D. (2000). Change of base in Bailey pairs. Ramanujan J., 4(4):435–453.
Chu, W. (2002). The Saalschütz chain reactions and bilateral basic hypergeometric series. Constructive Approx., 18:579–598.
Garrett, T., Ismail, M., and Stanton, D. (1999). Variants of the Rogers-Ramanujan identities. Adv. Appl. Math., 23(3):274–299.
Gessel, I. and Stanton, D. (1983). Applications of q-Lagrange inversion to basic hypergeometric series. Trans. Amer. Math. Soc., 277:173–201.
Milne, S. C. (1980). A multiple series transformation of the very well poised 4+2kΨ2k+4. Pacific J. Math., 91:419–430.
Schilling, A. and Warnaar, S. O. (1997). A higher-level Bailey lemma: Proof and application. Preprint.
Slater, L. J. (1951). A new proof of Rogers' transformations of infinite series. Proc. London Math. Soc. (2), 53:460–475.
Slater, L. J. (1952). Further identities of the Rogers-Ramanujan type. Proc. London Math. Soc. (2), 54:147–167.
Slater, L. J. (1966). Generalized Hypergeometric Functions. Cambridge University Press, Cambridge.
Stanton, D. (2001). The Bailey-Rogers-Ramanujan group, volume 291 of Contemporary Mathematics, pages 55–70. Amer. Math. Soc., Providence, RI.
Stembridge, J. R. (1990). Hall-Littlewood functions, plane partitions, and the Rogers-Ramanujan identities. Trans. Amer. Math. Soc., 319:469–498.
Warnaar, S. O. (2001). The generalized Borwein conjecture I: The Burge transform, volume 291 of Contemporary Mathematics, pages 243–267. Amer. Math. Soc., Providence, RI.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2005 Springer Science+Business Media, Inc.
About this chapter
Cite this chapter
Wenchang, C.H.U. (2005). The Saalschütz Chain Reactions and Multiple q-Series Transformations. In: Ismail, M.E., Koelink, E. (eds) Theory and Applications of Special Functions. Developments in Mathematics, vol 13. Springer, Boston, MA. https://doi.org/10.1007/0-387-24233-3_6
Download citation
DOI: https://doi.org/10.1007/0-387-24233-3_6
Publisher Name: Springer, Boston, MA
Print ISBN: 978-0-387-24231-6
Online ISBN: 978-0-387-24233-0
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)