Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 5 (2009), 050, 19 pages      arXiv:0904.3453      https://doi.org/10.3842/SIGMA.2009.050

Partial Sums of Two Quartic q-Series

Wenchang Chu a and Chenying Wang b
a) Dipartimento di Matematica, Università degli Studi di Salento, Lecce-Arnesano P. O. Box 193, Lecce 73100, Italy
b) College of Mathematics and Physics, Nanjing University of Information Science and Technology, Nanjing 210044, P. R. China

Received January 20, 2009, in final form April 17, 2009; Published online April 22, 2009

Abstract
The partial sums of two quartic basic hypergeometric series are investigated by means of the modified Abel lemma on summation by parts. Several summation and transformation formulae are consequently established.

Key words: basic hypergeometric series (q-series); well-poised q-series; quadratic q-series; cubic q-series; quartic q-series; the modified Abel lemma on summation by parts.

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References

  1. Andrews G.E., Three aspects of partitions, Séminaire Lotharingien de Combinatoire (Salzburg, 1990), Publ. Inst. Rech. Math. Av., Vol. 462, Univ. Louis Pasteur, Strasbourg, 1991, 5-18.
  2. Andrews G.E., Askey R., Roy R., Special functions, Encyclopedia of Mathematics and Its Applications, Vol. 71, Cambridge University Press, Cambridge, 1999.
  3. Bailey W.N., Well-poised basic hypergeometric series, Quart. J. Math. Oxford Ser. 18 (1947), 157-166.
  4. Chu W., Inversion techniques and combinatorial identities: Jackson's q-analogue of the Dougall-Dixon theorem and the dual formulae, Compositio Math. 95 (1995), 43-68.
  5. Chu W., Abel's lemma on summation by parts and basic hypergeometric series, Adv. in Appl. Math. 39 (2007), 490-514.
  6. Chu W., Jia C., Abel's method on summation by parts and theta hypergeometric series, J. Combin. Theory Ser. A 115 (2008), 815-844.
  7. Chu W., Wang C., Abel's lemma on summation by parts and partial q-series transformations, Sci. China Ser. A 52 (2009), 720-748.
  8. Chu W., Wang X., Abel's lemma on summation by parts and terminating q-series identities, Numer. Algorithms 49 (2008), 105-128.
  9. Frenkel I.B., Turaev V.G., Elliptic solutions of the Yang-Baxter equation and modular hypergeometric functions, The Arnold-Gelfand Mathematical Seminars, Birkhäuser Boston, Boston, MA, 1997, 171-204.
  10. Gasper G., Summation, transformation, and expansion formulas for bibasic series, Trans. Amer. Math. Soc. 312 (1989), 257-277.
  11. Gasper G., Rahman M., An indefinite bibasic summation formula and some quadratic, cubic and quartic summation and transformation formulas, Canad. J. Math. 42 (1990), 1-27.
  12. Gasper G., Rahman M., Basic hypergeometric series, 2nd ed., Encyclopedia of Mathematics and Its Applications, Vol. 96, Cambridge University Press, Cambridge, 2004.
  13. Gessel I., Stanton D., Applications of q-Lagrange inversion to basic hypergeometric series, Trans. Amer. Math. Soc. 277 (1983), 173-201.
  14. Ismail M.E.H., Stanton D., Tribasic integrals and identities of Rogers-Ramanujan type, Trans. Amer. Math. Soc. 355 (2003), 4061-4091.
  15. Koornwinder T.H., Askey-Wilson polynomials as zonal spherical functions on the SU(2) quantum group, SIAM J. Math. Anal. 24 (1993), 795-813.
  16. Rahman M., Some quadratic and cubic summation formulas for basic hypergeometric series, Canad. J. Math. 45 (1993), 394-411.
  17. Rahman M., Verma A., Quadratic transformation formulas for basic hypergeometric series, Trans. Amer. Math. Soc. 335 (1993), 277-302.
  18. Stanton D., The Bailey-Rogers-Ramanujan group, Contemp. Math. 291 (2001), 55-70.
  19. Stromberg K.R., An introduction to classical real analysis, Wadsworth International Mathematics Series, Wadsworth International, Belmont, Calif., 1981.
  20. Warnaar S.O., Summation and transformation formulas for elliptic hypergeometric series, Constr. Approx. 18 (2002), 479-502, math.QA/0001006.


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