Q-analogues of five difficult hypergeometric evaluations

A nonterminating balanced q -series is examined by means of the modified Abel lemma on summation by parts that leads to q -analogues of five difficult identities for classical hypergeometric series, including three formulae conjectured by Gosper in 1977

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  • [1] Andrews GE. Connection coefficient problems and partitions. Proceedings of Symposia in Pure Mathematics 1979; 34: 1-24.
  • [2] Bailey WN. Generalized Hypergeometric Series. Cambridge, UK: Cambridge University Press, 1935.
  • [3] Chen XJ, Chu W. Summation formulae for a class of terminating balanced q -series. Journal of Mathematical Analysis and Applications 2017; 451 (1): 508-523.
  • [4] Chu W. Abel’s lemma on summation by parts and basic hypergeometric series. Advances in Applied Mathematics 2007; 39 (4): 490-514.
  • [5] Chu W. Asymptotic method for Dougall’s bilateral hypergeometric sums. Bulletin des Sciences Mathématiques 2007; 131 (5): 457-468.
  • [6] Chu W. q -extensions of Dougall’s bilateral 2H2 -series. The Ramanujan Journal 2011; 25 (1): 121-139.
  • [7] Chu W. Evaluation of nonterminating 3F2( 3 4 )-series. Journal of Mathematical Analysis and Applications 2017; 450 (1): 490-503.
  • [8] Chu W. Evaluating a class of balanced q -series. Turkish Journal of Mathematics 2018; 42(5): 2699-2706.
  • [9] Gasper G, Rahman M. Basic Hypergeometric Series (2nd ed.). Cambridge, UK: Cambridge University Press, 2004.
  • [10] Gessel I, Stanton D. Applications of q -Lagrange inversion to basic hypergeometric series. Transactions of the American Mathematical Society 1983; 277: 173-201.
  • [11] Gradshteyn IS, Ryzhik IM. Table of Integrals, Series and Products (sixth edition). Singapore: Elsevier, 2007.
  • [12] Rainville ED. Special Functions. New York, USA: The Macmillan Company, 1960.
  • [13] Wang CY, Chen XJ. A short proof for Gosper’s 7F6 -series conjecture. Journal of Mathematical Analysis and Applications 2015; 422 (2): 819-824.
  • [14] Wang CY, Chen XJ. New proof for a nonterminating cubic hypergeometric series identiy of Gasper–Rahman. Journal of Nanjing University Mathematical Biquarterly 2015; 32 (1): 38-45.
  • [15] Wang CY, Dai JJ, Mező I. A nonterminating 7F6 -series evaluation. Integral Transforms and Special Functions 2018; 29 (9): 719-724.