Binomial sums via Bailey's cubic transformation

Wenchang Chu

Czechoslovak Mathematical Journal (2023)

  • Volume: 73, Issue: 4, page 1131-1150
  • ISSN: 0011-4642

Abstract

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By employing one of the cubic transformations (due to W. N. Bailey (1928)) for the 3 F 2 ( x ) -series, we examine a class of 3 F 2 ( 4 ) -series. Several closed formulae are established by means of differentiation, integration and contiguous relations. As applications, some remarkable binomial sums are explicitly evaluated, including one proposed recently as an open problem.

How to cite

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Chu, Wenchang. "Binomial sums via Bailey's cubic transformation." Czechoslovak Mathematical Journal 73.4 (2023): 1131-1150. <http://eudml.org/doc/299558>.

@article{Chu2023,
abstract = {By employing one of the cubic transformations (due to W. N. Bailey (1928)) for the $_3F_2(x)$-series, we examine a class of $_3F_2(4)$-series. Several closed formulae are established by means of differentiation, integration and contiguous relations. As applications, some remarkable binomial sums are explicitly evaluated, including one proposed recently as an open problem.},
author = {Chu, Wenchang},
journal = {Czechoslovak Mathematical Journal},
keywords = {hypergeometric series; Bailey's cubic transformation; contiguous relation; reversal series; binomial coefficient},
language = {eng},
number = {4},
pages = {1131-1150},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Binomial sums via Bailey's cubic transformation},
url = {http://eudml.org/doc/299558},
volume = {73},
year = {2023},
}

TY - JOUR
AU - Chu, Wenchang
TI - Binomial sums via Bailey's cubic transformation
JO - Czechoslovak Mathematical Journal
PY - 2023
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 73
IS - 4
SP - 1131
EP - 1150
AB - By employing one of the cubic transformations (due to W. N. Bailey (1928)) for the $_3F_2(x)$-series, we examine a class of $_3F_2(4)$-series. Several closed formulae are established by means of differentiation, integration and contiguous relations. As applications, some remarkable binomial sums are explicitly evaluated, including one proposed recently as an open problem.
LA - eng
KW - hypergeometric series; Bailey's cubic transformation; contiguous relation; reversal series; binomial coefficient
UR - http://eudml.org/doc/299558
ER -

References

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  1. Bailey, W. N., 10.1112/plms/s2-28.1.242, Proc. Lond. Math. Soc. (2) 28 (1928), 242-254 9999JFM99999 54.0392.04. (1928) MR1575853DOI10.1112/plms/s2-28.1.242
  2. Bailey, W. N., Generalized Hypergeometric Series, Cambridge Tracts in Mathematics and Mathematical Physics 32. Cambridge University Press, Cambridge (1935). (1935) Zbl0011.02303MR0185155
  3. Campbell, J. M., Solution to a problem due to Chu and Kiliç, Integers 22 (2022), Article ID A46, 8 pages. (2022) Zbl1493.11044MR4430897
  4. Chen, X., Chu, W., 10.1080/10652469.2017.1376194, Integral Transform Spec. Funct. 28 (2017), 825-837. (2017) Zbl1379.33012MR3724507DOI10.1080/10652469.2017.1376194
  5. Chu, W., Inversion techniques and combinatorial identities: A quick introduction to hypergeometric evaluations, Runs and Patterns in Probability Mathematics and its Applications 283. Kluwer, Dordrecht (1994), 31-57. (1994) Zbl0830.05006MR1292876
  6. Chu, W., 10.1216/rmjm/1030539687, Rocky Mt. J. Math. 32 (2002), 561-587. (2002) Zbl1038.33002MR1934906DOI10.1216/rmjm/1030539687
  7. Chu, W., 10.1090/proc/13293, Proc. Am. Math. Soc. 145 (2017), 1031-1040. (2017) Zbl1352.33001MR3589303DOI10.1090/proc/13293
  8. Chu, W., 10.1016/j.disc.2018.07.028, Discrete Math. 341 (2018), 3159-3164. (2018) Zbl1395.05024MR3854125DOI10.1016/j.disc.2018.07.028
  9. Chu, W., 10.1007/s00574-021-00252-x, Bull. Braz. Math. Soc. (N.S.) 53 (2022), 95-105. (2022) Zbl1484.05012MR4379425DOI10.1007/s00574-021-00252-x
  10. Chu, W., Kiliç, E., 10.1216/rmj.2021.51.1221, Rocky Mt. J. Math. 51 (2021), 1221-1225. (2021) Zbl1473.05017MR4298842DOI10.1216/rmj.2021.51.1221
  11. Gessel, I. M., 10.1006/jsco.1995.1064, J. Symb. Comput. 20 (1995), 537-566. (1995) Zbl0908.33004MR1395413DOI10.1006/jsco.1995.1064
  12. Gessel, I. M., Stanton, D., 10.1137/0513021, SIAM J. Math. Anal. 13 (1982), 295-308. (1982) Zbl0486.33003MR0647127DOI10.1137/0513021
  13. Mikić, J., Two new identities involving the Catalan numbers and sign-reversing involutions, J. Integer Seq. 22 (2019), Article ID 19.7.7, 10 pages. (2019) Zbl1431.05020MR4040982
  14. Zeilberger, D., Forty ``strange" computer-discovered and computer-proved (of course) hypergeometric series evaluations, Available at (2004). (2004) 
  15. Zhou, R. R., Chu, W., 10.1007/s00373-016-1694-y, Graphs Comb. 32 (2016), 2183-2197. (2016) Zbl1351.05034MR3543225DOI10.1007/s00373-016-1694-y

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