Abstract
In terms of Dougall’s \(_2H_2\) series identity and the series rearrangement method, we establish a symmetric formula for hypergeometric series. Then it is utilized to derive a known nonterminating form of Saalschütz’s theorem. Similarly, we also show that Bailey’s \(_6\psi _6\) series identity implies the nonterminating form of Jackson’s \(_8\phi _7\) summation formula. Considering the reversibility of the proofs, it is routine to show that Dougall’s \(_2H_2\) series identity is equivalent to a known nonterminating form of Saalschütz’s theorem and Bailey’s \(_6\psi _6\) series identity is equivalent to the nonterminating form of Jackson’s \(_8\phi _7\) summation formula.
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References
Andrews, G.E.: Ramanujan’s “lost” notebook. I. Partial \(\theta \)-function. Adv. Math. 41, 137–172 (1981)
Andrews, G.E., Askey, R., Roy, R.: Special Functions. Cambridge University Press, Cambridge (2000)
Chen, W.Y.C., Fu, A.M.: Semi-finite forms of bilateral basic hypergeometric series. Prop. Am. Math. Soc. 134, 1719–1725 (2006)
Gasper, G., Rahman, M.: Basic Hypergeometric Series, 2nd edn. Cambridge University Press, Cambridge (2004)
Guo, V.J.W.: A \(q\)-analogue of the (I.2) supercongruence of Van Hamme. Int. J. Number Theory 15, 29–36 (2019)
Guo, V.J.W., Zudilin, W.: A \(q\)-microscope for supercongruences. Adv. Math. 346, 329–358 (2019)
Jouhet, F.: Some more semi-finite forms of bilateral basic hypergeometric series. Ann. Comb. 11, 47–57 (2007)
Kang, S.-Y.: Generalizations of Ramanujan’s reciprocity theorem and their applications. J. Lond. Math. Soc. 2(75), 18–34 (2007)
Liu, Z.-G.: Some operator identities and \(q\)-series transformation formulas. Discret. Math. 265, 119–139 (2003)
Liu, Z.-G.: Extensions of Ramanujan’s reciprocity theorem and the Andrews–Askey integral. J. Math. Anal. Appl. 443, 1110–1229 (2016)
Ma, X.R.: Six-variable generalization of Ramanujan’s reciprocity theorem. J. Math. Anal. Appl. 353, 320–328 (2009)
Slater, L.J.: Generalized Hypergeometric Functions. Cambridge University Press, Cambridge (1966)
Wei, C., Yan, Q., Li, J.: A short proof for Dougall’s \(_2H_2\)-series identity. Discret. Math. 312, 2997–2999 (2012)
Zhang, C., Zhang, Z.: Two new transformation formulas of basic hypergeometric series. J. Math. Anal. Appl. 336, 777–787 (2007)
Zhang, Z., Hu, Q.: On the very-well-poised bilateral basic hypergeometric \(_7\psi _7\) series. J. Math. Anal. Appl. 367, 657–668 (2010)
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The work is supported by the National Natural Science Foundation of China (No. 11661032).
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Wei, C. A symmetric formula for hypergeometric series. Ramanujan J 55, 919–927 (2021). https://doi.org/10.1007/s11139-019-00248-8
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DOI: https://doi.org/10.1007/s11139-019-00248-8
Keywords
- Hypergeometric series
- Dougall’s \(_2H_2\)-series identity
- Basic Hypergeometric series
- Bailey’s \(_6\psi _6\) series identity