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Articles

General q-series transformations based on Abel's lemma on summation by parts and their applications

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Pages 553-576 | Received 13 Jan 2023, Accepted 08 Dec 2023, Published online: 19 Jan 2024

References

  • G. Bhatnagar, In praise of an elementary identity of Euler, Electron. J. Combin. 18 (2011), pp. #13.
  • G. Bhatnagar and S. Milne, Generalized bibasic hypergeometric series and their U(n) extensions, Adv. Math. 131 (1997), pp. 188–252.
  • K.X. Bian, Y. Liu, and Y.P. Mu, q-Gosper algorithm and simple Bailey pairs, J. Symbolic Comput.113 (2022), pp. 39–52.
  • J.M. Campbell and M. Cantarini, A series evaluation technique based on a modified Abel lemma, Turkish J. Math. 46 (2022), pp. 1520–1537.
  • V.Y.B. Chen, W.Y.C. Chen, and N.S.S. Gu, The Abel lemma and the q-Gosper algorithm, Math. Comput. 77 (2008), pp. 1057–1074.
  • W.Y.C. Chen, Q.H. Hou, and H.T. Jin, The Abel-Zeilberger algorithm, Electron. J. Combin. 18 (2011), pp. #17.
  • K.S. Chong and K. Lam, A generalized Abel's partial summation formula and its application in self-organizing systems, Stoc. Models 15 (1999), pp. 779–790.
  • W.C. Chu, Abel's method on summation by parts and hypergeometric series, J. Diff. Equ. Appli. 12 (2006), pp. 783–798.
  • W.C. Chu, Bailey's very well-poised 6ψ6-series identity, J. Combin. Theory Ser. A 113 (2006), pp. 966–979.
  • W.C. Chu, Abel's lemma on summation by parts and basic hypergeometric series, Adv. Appl. Math.39 (2007), pp. 490–514.
  • W.C. Chu, Summation formulae for quintic q-series, Mathematics 10 (2022), pp. 2210. https://doi.org/10.3390/math10132210.
  • W.C. Chu and C.Z. Jia, Abel's method on summation by parts and theta hypergeometric series, J. Combin. Theory Ser. A 115 (2008), pp. 815–844.
  • W.C. Chu and C.Y. Wang, Abel's lemma on summation by parts and partial q-series transformations, Sci. China Ser. A 52 (2009), pp. 720–748.
  • G. Gasper, Summation, transformation, and expansion formulas for bibasic series, Trans. Amer. Math. Soc. 312 (1989), pp. 257–277.
  • G. Gasper and M. Rahman, An indefinite bibasic summation formula and some quadratic, cubic, and quartic summation and transformation formulas, Canad. J. Math. 42 (1990), pp. 1–27.
  • G. Gasper and M. Rahman, Basic Hypergeometric Series, 2nd ed., Cambridge University Press, Cambridge, 2004.
  • V.K. Jain and A. Verma, Summation formulae for bibasic and polybasic hypergeometric series, Indian Math. Soc. 63 (1997), pp. 211–223.
  • H.T. Jin and L.H. Sun, On Spieß's conjecture on harmonic numbers, Discrete Appl. Math. 161 (2013), pp. 2038–2041.
  • K. Knopp, Theory and Application of Infinite Series, Dover Books on Mathematics, Dover Publications, Mineola, NY, 1990.
  • T. Komatsu and R.S. Li, Infinite series containing generalized q-harmonic numbers, Integers 21 (2021), Article ID 13pp.
  • T.H. Koornwinder, On the equivalence of two fundamental theta identities, Anal. Appl. (Singap.) 12 (2014), pp. 711–725.
  • Z.G. Liu, A three-term theta function identity and its applications, Adv. Math. 195 (2005), pp. 1–23.
  • Z.G. Liu, Addition formulas for Jacobi theta functions, Dedekind's eta function, and Ramanujan's congruences, Pacific J. Math. 240 (2009), pp. 135–150.
  • X.R. Ma, The (f,g)-inversion formula and its applications: the (f,g)-summation formula, Adv. Appl. Math. 38 (2007), pp. 227–257.
  • M. Rahman and A. Verma, Quadratic transformation formulas for basic hypergeometric series, Trans. Amer. Math. Soc. 335 (1993), pp. 277–302.
  • D. Roselin, Bilateral representation of Gasper and Rahman's summation formulae, Proc. Nat. Acad. Sci. Indian. 80A (2010), pp. 145–148.
  • H. Rosengren and M. Schlosser, Multidimensional matrix inversions and elliptic hypergeometric series on root systems, SIGMA 16 (2020), Article ID 21 pp.
  • J. Wang, A new elliptic interpolation formula via the (f,g)-inversion, Proc. Amer. Math. Soc. 148 (2020), pp. 3457–3471.
  • C.Y. Wang and J.N. Xu, Three cubic q-series of Gasper and Rahman, Ramanujan J. 59 (2022), pp. 199–210. https://doi.org/10.1007/s11139-021-00502-y.
  • C.Y. Wang, J.J. Dai, and I. Mezö, A nonterminating 7F6-series evaluation, Integral Transforms Spec. Funct. 29 (2018), pp. 719–724.
  • S.O. Warnaar, Summation and transformation formulas for elliptic hypergeometric series, Constr. Approx. 18 (2002), pp. 479–502.
  • J.N. Xu and X.R. Ma, General q-series transformations based on Abel's lemma on summation by parts and their applications (extended version), 32pp. Available at https://arxiv.org/abs/2307.07968.
  • Y.S. Zhang and T.M. Wang, Some applications of Gasper's bibasic summation formula, Rocky Mountain J. Math. 38 (2008), pp. 703–712.
  • Z. Zhang and Y. Zhang, Summation formulas of q-series by modified Abel's lemma, Adv. Stud. Contemp. Math. 17 (2008), pp. 119–129.

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