Year 2020, Volume 49 , Issue 5, Pages 1611 - 1624 2020-10-06

Nonterminating well–poised hypergeometric series

Wenchang CHU [1]


Two classes of nonterminating well--poised series are examined by means of the modified Abel lemma on summation by parts, that leads to several summation and transformation formulae.

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Classical hypergeometric series, Abel's lemma on summation by parts, Pfaff–Saalschütz theorem, Dougall's formulae for $_2H_2$-series and $_5H_5$-series
  • [1] W.N. Bailey, Generalized Hypergeometric Series, Cambridge University Press, Cambridge, 1935.
  • [2] W.N. Bailey, Series of hypergeometric type which are infinite in both directions, Quart. J. Math. 7, 105–115, 1936.
  • [3] T.J.I’A. Bromwich, An Introduction to the Theory of Infinite Series (2nd ed), Macmillan, New York, 1959.
  • [4] X.J. Chen and W. Chu, Closed formulae for a class of terminating $_3F_2(4)$-series, Integral Transforms Spec. Funct. 28 (11), 825–837, 2017.
  • [5] X.J. Chen and W. Chu, Terminating $_3F_2(4)$-series extended with three integer parameters, J. Difference Equ. Appl. 24 (8), 1346–1367, 2018.
  • [6] W. Chu, Abel’s lemma on summation by parts and Ramanujan’s $_1\psi_1$-series Identity, Aequationes Math. 72 (1-2), 172–176, 2006.
  • [7] W. Chu, Abel’s method on summation by parts and hypergeometric series, J. Difference Equ. Appl. 12 (8), 783–798, 2006.
  • [8] W. Chu, Bailey’s very well–poised ${_6\psi_6}$-series identity, J. Combin. Theory Ser. 113 (6), 966–979, 2006.
  • [9] W. Chu, Abel’s lemma on summation by parts and basic hypergeometric series, Adv. Appl. Math. 39 (4), 490–514, 2007.
  • [10] W. Chu, Asymptotic method for Dougall’s bilateral hypergeometric sums, Bull. Sci. Math. 131 (5), 457–468, 2007.
  • [11] W. Chu, q-extensions of Dougall’s bilateral ${_2H_2}$-series, Ramanujan J. 25 (1), 121–139, 2011.
  • [12] W. Chu, Evaluation of nonterminating hypergeometric $_3F_2(\frac34)$-series, J. Math. Anal. Appl. 450 (1), 490–503, 2017.
  • [13] W. Chu and X. Wang, The modified Abel lemma on summation by parts and terminating hypergeometric series identities, Integral Transforms Spec. Funct. 20 (2), 93–118, 2009.
  • [14] W. Chu, X. Wang, and D.Y. Zheng, Application of the residue theorem to bilateral hypergeometric series, Matematiche 62 (2), 127–146, 2007.
  • [15] A.C. Dixon, Summation of a certain series, Proc. London Math. Soc. 35 (1), 284–291, 1903.
  • [16] M.A. Dougall, On Vandermonde’s theorem and some more general expansion, Proc. Edin. Math. Soc. 25, 114–132, 2007.
  • [17] R.L. Graham, D.E. Knuth, and O. Patashnik, Concrete Mathematics, Addison-Wesley Publ. Company, Reading, Massachusetts, 1989.
  • [18] M. Jackson, A note on the sum of a particular well–poised $_6H_6$ with argument −1, J. London Math. Soc. 27, 124–126, 1952.
  • [19] K. Knopp, Theory and Applications of Infinite Series, Hafner Publishing Company, New York, 1971.
  • [20] E.D. Rainville, Special Functions, New York, The Macmillan Company, 1960.
  • [21] H.M. Srivastava, Y. Vyas, and K. Fatawat, Extensions of the classical theorems for very well–poised hypergeometric functions, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 113 (367), 2019, https://doi.org/10.1007/s13398-017-0485-5.
  • [22] K.R. Stromberg, An Introduction to Classical Real Analysis, Wadsworth, INC. Belmont, California, 1981.
  • [23] C. Wang and X. Chen, A short proof for Gosper’s $_7F_6$-series conjecture, J. Math. Anal. Appl. 422 (2), 819–824, 2015.
  • [24] C.Wang, J. Dai, and I. Mezo, A nonterminating $_7F_6$-series evaluation, Integral Transforms Spec. Funct. 29 (9), 719–724, 2018.
  • [25] F.J.W. Whipple, On well–poised series, generalized hypergeometric series having parameters in pairs, each pair with the same sum, Proc. London Math. Soc. 24 (2), 247–263, 1926.
Primary Language en
Subjects Mathematics
Journal Section Mathematics
Authors

Orcid: 0000-0002-8425-212X
Author: Wenchang CHU (Primary Author)
Institution: Zhoukou Normal University
Country: China


Dates

Publication Date : October 6, 2020

Bibtex @research article { hujms548103, journal = {Hacettepe Journal of Mathematics and Statistics}, issn = {2651-477X}, eissn = {2651-477X}, address = {}, publisher = {Hacettepe University}, year = {2020}, volume = {49}, pages = {1611 - 1624}, doi = {10.15672/hujms.548103}, title = {Nonterminating well–poised hypergeometric series}, key = {cite}, author = {Chu, Wenchang} }
APA Chu, W . (2020). Nonterminating well–poised hypergeometric series . Hacettepe Journal of Mathematics and Statistics , 49 (5) , 1611-1624 . DOI: 10.15672/hujms.548103
MLA Chu, W . "Nonterminating well–poised hypergeometric series" . Hacettepe Journal of Mathematics and Statistics 49 (2020 ): 1611-1624 <https://dergipark.org.tr/en/pub/hujms/issue/57199/548103>
Chicago Chu, W . "Nonterminating well–poised hypergeometric series". Hacettepe Journal of Mathematics and Statistics 49 (2020 ): 1611-1624
RIS TY - JOUR T1 - Nonterminating well–poised hypergeometric series AU - Wenchang Chu Y1 - 2020 PY - 2020 N1 - doi: 10.15672/hujms.548103 DO - 10.15672/hujms.548103 T2 - Hacettepe Journal of Mathematics and Statistics JF - Journal JO - JOR SP - 1611 EP - 1624 VL - 49 IS - 5 SN - 2651-477X-2651-477X M3 - doi: 10.15672/hujms.548103 UR - https://doi.org/10.15672/hujms.548103 Y2 - 2019 ER -
EndNote %0 Hacettepe Journal of Mathematics and Statistics Nonterminating well–poised hypergeometric series %A Wenchang Chu %T Nonterminating well–poised hypergeometric series %D 2020 %J Hacettepe Journal of Mathematics and Statistics %P 2651-477X-2651-477X %V 49 %N 5 %R doi: 10.15672/hujms.548103 %U 10.15672/hujms.548103
ISNAD Chu, Wenchang . "Nonterminating well–poised hypergeometric series". Hacettepe Journal of Mathematics and Statistics 49 / 5 (October 2020): 1611-1624 . https://doi.org/10.15672/hujms.548103
AMA Chu W . Nonterminating well–poised hypergeometric series. Hacettepe Journal of Mathematics and Statistics. 2020; 49(5): 1611-1624.
Vancouver Chu W . Nonterminating well–poised hypergeometric series. Hacettepe Journal of Mathematics and Statistics. 2020; 49(5): 1611-1624.