Geodesic
Binomial sums via Bailey's cubic transformation
Czechoslovak Mathematical Journal, Tome 73 (2023) no. 4, pp. 1131-1150
Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

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.
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.
DOI : 10.21136/CMJ.2023.0429-22
Classification : 05A19, 11B65, 33C20
Keywords: hypergeometric series; Bailey's cubic transformation; contiguous relation; reversal series; binomial coefficient 
@article{10_21136_CMJ_2023_0429_22,
     author = {Chu, Wenchang},
     title = {Binomial sums via {Bailey's} cubic transformation},
     journal = {Czechoslovak Mathematical Journal},
     pages = {1131--1150},
     year = {2023},
     volume = {73},
     number = {4},
     doi = {10.21136/CMJ.2023.0429-22},
     zbl = {07790565},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.21136/CMJ.2023.0429-22/}
}
TY  - JOUR
AU  - Chu, Wenchang
TI  - Binomial sums via Bailey's cubic transformation
JO  - Czechoslovak Mathematical Journal
PY  - 2023
SP  - 1131
EP  - 1150
VL  - 73
IS  - 4
UR  - http://geodesic.mathdoc.fr/articles/10.21136/CMJ.2023.0429-22/
DO  - 10.21136/CMJ.2023.0429-22
LA  - en
ID  - 10_21136_CMJ_2023_0429_22
ER  - 
%0 Journal Article
%A Chu, Wenchang
%T Binomial sums via Bailey's cubic transformation
%J Czechoslovak Mathematical Journal
%D 2023
%P 1131-1150
%V 73
%N 4
%U http://geodesic.mathdoc.fr/articles/10.21136/CMJ.2023.0429-22/
%R 10.21136/CMJ.2023.0429-22
%G en
%F 10_21136_CMJ_2023_0429_22
Chu, Wenchang. Binomial sums via Bailey's cubic transformation. Czechoslovak Mathematical Journal, Tome 73 (2023) no. 4, pp. 1131-1150. doi: 10.21136/CMJ.2023.0429-22

[1] Bailey, W. N.: Products of generalized hypergeometric series. Proc. Lond. Math. Soc. (2) 28 (1928), 242-254 \99999JFM99999 54.0392.04. | DOI | MR

[2] Bailey, W. N.: Generalized Hypergeometric Series. Cambridge Tracts in Mathematics and Mathematical Physics 32. Cambridge University Press, Cambridge (1935). | MR | JFM

[3] Campbell, J. M.: Solution to a problem due to Chu and Kiliç. Integers 22 (2022), Article ID A46, 8 pages. | MR | JFM

[4] Chen, X., Chu, W.: Closed formulae for a class of terminating $_3F_2(4)$-series. Integral Transform Spec. Funct. 28 (2017), 825-837. | DOI | MR | JFM

[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. | MR | JFM

[6] Chu, W.: Inversion techniques and combinatorial identities: Balanced hypergeometric series. Rocky Mt. J. Math. 32 (2002), 561-587. | DOI | MR | JFM

[7] Chu, W.: Terminating $_2F_1(4)$-series perturbed by two integer parameters. Proc. Am. Math. Soc. 145 (2017), 1031-1040. | DOI | MR | JFM

[8] Chu, W.: Further identities on Catalan numbers. Discrete Math. 341 (2018), 3159-3164. | DOI | MR | JFM

[9] Chu, W.: Alternating convolutions of Catalan numbers. Bull. Braz. Math. Soc. (N.S.) 53 (2022), 95-105. | DOI | MR | JFM

[10] Chu, W., Kiliç, E.: Binomial sums involving Catalan numbers. Rocky Mt. J. Math. 51 (2021), 1221-1225. | DOI | MR | JFM

[11] Gessel, I. M.: Finding identities with the WZ method. J. Symb. Comput. 20 (1995), 537-566. | DOI | MR | JFM

[12] Gessel, I. M., Stanton, D.: Strange evaluations of hypergeometric series. SIAM J. Math. Anal. 13 (1982), 295-308. | DOI | MR | JFM

[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. | MR | JFM

[14] Zeilberger, D.: Forty ``strange" computer-discovered and computer-proved (of course) hypergeometric series evaluations. Available at {\def\let \relax \brokenlink{ https://sites.math.rutgers.edu/ zeilberg/mamarim/mamarimhtml/strange.html}}\kern0pt (2004).

[15] Zhou, R. R., Chu, W.: Identities on extended Catalan numbers and their $q$-analogs. Graphs Comb. 32 (2016), 2183-2197. | DOI | MR | JFM

Cité par Sources :