Rocky Mountain Journal of Mathematics

Carlitz inversions and identities of the Rogers-Ramanujan type

Xiaojing Chen and Wenchang Chu

Full-text: Open access

Abstract

By means of the inverse series relations due to Carlitz \cite{carlitz}, we establish several transformation formulae for nonterminating $q$-series, which will systematically be employed to review identities of the Rogers-Ramanujan type moduli 5, 7, 8, 10, 14 and 27.

Article information

Source
Rocky Mountain J. Math., Volume 44, Number 4 (2014), 1125-1142.

Dates
First available in Project Euclid: 31 October 2014

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1414760945

Digital Object Identifier
doi:10.1216/RMJ-2014-44-4-1125

Mathematical Reviews number (MathSciNet)
MR3274340

Zentralblatt MATH identifier
1375.33029

Subjects
Primary: 33D15: Basic hypergeometric functions in one variable, $_r\phi_s$
Secondary: 05A30: $q$-calculus and related topics [See also 33Dxx]

Keywords
Basic hypergeometric series Rogers-Ramanujan identities

Citation

Chen, Xiaojing; Chu, Wenchang. Carlitz inversions and identities of the Rogers-Ramanujan type. Rocky Mountain J. Math. 44 (2014), no. 4, 1125--1142. doi:10.1216/RMJ-2014-44-4-1125. https://projecteuclid.org/euclid.rmjm/1414760945


Export citation

References

  • A.K. Agarwal and D.M. Bressoud, Lattice paths and multiple basic hypergeometric series, Pacific J. Math. 136 (1989), 209–228.
  • G.E. Andrews, On the $q$-analog of Kummer's theorem and applications, Duke Math. J. 40 (1973), 525–528.
  • ––––, On $q$-analogues of the Watson and Whipple summations, SIAM. J. Math. Anal. 7 (1976), 332–336.
  • ––––, Ramanujan's “lost" notebook: II. $\vartheta $-function expansions, Adv. Math. 41 (1981), 173–185.
  • ––––, Combinatorics and Ramanujan's “lost" notebook, in Surveys in combinatorics, Lect. Note 103, London Mathematical Society, 1985.
  • ––––, $q$-series: Their development and application in analysis, number theory, combinatorics, physics, and computer algebra, CBMS Conf. Ser. 66, American Mathematical Society, Providence, RI, 1986.
  • W.N. Bailey, Generalized hypergeometric series, Cambridge University Press, Cambridge, 1935.
  • W.N. Bailey, Some identities in combinatory analysis, Proc. Lond. Math. Soc. 49 (1947), 421–435.
  • ––––, Identities of the Rogers-Ramanujan type, Proc. Lond. Math. Soc. 50 (1948), 1–10.
  • ––––, On the simplification of some identities of the Rogers-Ramanujan type, Proc. Lond. Math. Soc. 1 (1951), 217–221.
  • L. Carlitz, Some inverse series relations, Duke Math. J. 40 (1973), 893–901.
  • W. Chu, Gould-Hus-Carlitz inversions and Rogers-Ramanujan identities, Acta Math. Sinica 331 (1990), 7–12.
  • ––––, Durfee rectangles and the Jacobi triple product identity, Acta Math. Sinica [New Series] 9 (1993), 24–26.
  • ––––, Inversion techniques and combinatorial identities, Boll. Un. Mat. Ital. 7 (1993), 737–760.
  • ––––, Inversion techniques and combinatorial identities: Strange evaluations of basic hypergeometric series, Comp. Math. 91 (1994), 121–144.
  • ––––, Basic hypergeometric identities: An introductory revisiting through the Carlitz inversions, Forum Math. 7 (1995), 117–129.
  • W. Chu and C.Y. Wang, The multisection method for triple products and identities of Rogers-Ramanujan type, J. Math. Anal. Appl. 339 (2008), 774–784.
  • G. Gasper and M. Rahman, Basic hypergeometric series, 2nd ed., Cambridge University Press, Cambridge, 2004.
  • I. Gessel and D. Stanton, Applications of $q$-Lagrange inversion to basic hypergeometric series, Trans. Amer. Math. Soc. 277 (1983), 173–201.
  • H.W. Gould and L.C. Hsu, Some new inverse series relations, Duke Math. J. 40 (1973), 885–891.
  • F.H. Jackson, Examples of a generalization of Euler's transformation for power series, Mess. Math. 57 (1928), 169–187.
  • C.G.J. Jacobi, Fundamenta Nova Theoriae Functionum Ellipticarum, Fratrum Bornträger Regiomonti, 1829; Gesammelte werke, Erster Band, G. Reimer, Berlin, 1881.
  • L.J. Rogers, Second memoir on the expansion of certain infinite products, Proc. Lond. Math. Soc. 25 (1894), 318–343.
  • ––––, On two theorems of combinatory analysis and some allied identities, Proc. Lond. Math. Soc. 16 (1917), 315–336.
  • A. Selberg, Über einige arithmetische identitäten, Avh. Norske Vidensk. Akad. Oslo Mat. Natur. 8 (1936), 2–23.
  • A.V. Sills, Identities of the Rogers-Ramanujan-Slater type, Int. J. Num. Theor. 3 (2007), 293–323.
  • L.J. Slater, Generalized hypergeometric functions, Cambridge University Press, Cambridge, 1966.
  • ––––, A new proof of Rogers's transformations of infinite series, Proc. Lond. Math. Soc. 53 (1951), 460–475.
  • L.J. Slater, Further identities of the Rogers-Ramanujan type, Proc. Lond. Math. Soc. 54 (1952), 147–167.
  • G.N. Watson, A new proof of the Rogers-Ramanujan identities, J. Lond. Math. Soc. 4 (1929), 4–9.