Article
| Title: | Binomial sums via Bailey's cubic transformation (English) |
| Author: | Chu, Wenchang |
| Language: | English |
| Journal: | Czechoslovak Mathematical Journal |
| ISSN: | 0011-4642 (print) |
| ISSN: | 1572-9141 (online) |
| Volume: | 73 |
| Issue: | 4 |
| Year: | 2023 |
| Pages: | 1131-1150 |
| Summary lang: | English |
| . | |
| Category: | math |
| . | |
| Summary: | 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. (English) |
| Keyword: | hypergeometric series |
| Keyword: | Bailey's cubic transformation |
| Keyword: | contiguous relation |
| Keyword: | reversal series |
| Keyword: | binomial coefficient |
| MSC: | 05A19 |
| MSC: | 11B65 |
| MSC: | 33C20 |
| DOI: | 10.21136/CMJ.2023.0429-22 |
| . | |
| Date available: | 2023-11-23T12:23:35Z |
| Last updated: | 2023-11-27 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/151951 |
| . | |
| Reference: | [1] Bailey, W. N.: Products of generalized hypergeometric series.Proc. Lond. Math. Soc. (2) 28 (1928), 242-254 \99999JFM99999 54.0392.04. MR 1575853, 10.1112/plms/s2-28.1.242 |
| Reference: | [2] Bailey, W. N.: Generalized Hypergeometric Series.Cambridge Tracts in Mathematics and Mathematical Physics 32. Cambridge University Press, Cambridge (1935). Zbl 0011.02303, MR 0185155 |
| Reference: | [3] Campbell, J. M.: Solution to a problem due to Chu and Kiliç.Integers 22 (2022), Article ID A46, 8 pages. Zbl 1493.11044, MR 4430897 |
| Reference: | [4] Chen, X., Chu, W.: Closed formulae for a class of terminating $_3F_2(4)$-series.Integral Transform Spec. Funct. 28 (2017), 825-837. Zbl 1379.33012, MR 3724507, 10.1080/10652469.2017.1376194 |
| Reference: | [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. Zbl 0830.05006, MR 1292876 |
| Reference: | [6] Chu, W.: Inversion techniques and combinatorial identities: Balanced hypergeometric series.Rocky Mt. J. Math. 32 (2002), 561-587. Zbl 1038.33002, MR 1934906, 10.1216/rmjm/1030539687 |
| Reference: | [7] Chu, W.: Terminating $_2F_1(4)$-series perturbed by two integer parameters.Proc. Am. Math. Soc. 145 (2017), 1031-1040. Zbl 1352.33001, MR 3589303, 10.1090/proc/13293 |
| Reference: | [8] Chu, W.: Further identities on Catalan numbers.Discrete Math. 341 (2018), 3159-3164. Zbl 1395.05024, MR 3854125, 10.1016/j.disc.2018.07.028 |
| Reference: | [9] Chu, W.: Alternating convolutions of Catalan numbers.Bull. Braz. Math. Soc. (N.S.) 53 (2022), 95-105. Zbl 1484.05012, MR 4379425, 10.1007/s00574-021-00252-x |
| Reference: | [10] Chu, W., Kiliç, E.: Binomial sums involving Catalan numbers.Rocky Mt. J. Math. 51 (2021), 1221-1225. Zbl 1473.05017, MR 4298842, 10.1216/rmj.2021.51.1221 |
| Reference: | [11] Gessel, I. M.: Finding identities with the WZ method.J. Symb. Comput. 20 (1995), 537-566. Zbl 0908.33004, MR 1395413, 10.1006/jsco.1995.1064 |
| Reference: | [12] Gessel, I. M., Stanton, D.: Strange evaluations of hypergeometric series.SIAM J. Math. Anal. 13 (1982), 295-308. Zbl 0486.33003, MR 0647127, 10.1137/0513021 |
| Reference: | [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. Zbl 1431.05020, MR 4040982 |
| Reference: | [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). |
| Reference: | [15] Zhou, R. R., Chu, W.: Identities on extended Catalan numbers and their $q$-analogs.Graphs Comb. 32 (2016), 2183-2197. Zbl 1351.05034, MR 3543225, 10.1007/s00373-016-1694-y |
| . |
Fulltext not available (moving wall 24 months)
Search
Partner of
