| Title:
|
Some finite generalizations of Euler's pentagonal number theorem (English) |
| Author:
|
Liu, Ji-Cai |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
67 |
| Issue:
|
2 |
| Year:
|
2017 |
| Pages:
|
525-531 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Euler's pentagonal number theorem was a spectacular achievement at the time of its discovery, and is still considered to be a beautiful result in number theory and combinatorics. In this paper, we obtain three new finite generalizations of Euler's pentagonal number theorem. (English) |
| Keyword:
|
$q$-binomial coefficient |
| Keyword:
|
$q$-binomial theorem |
| Keyword:
|
pentagonal number theorem |
| MSC:
|
05A17 |
| MSC:
|
11B65 |
| idZBL:
|
Zbl 06738535 |
| idMR:
|
MR3661057 |
| DOI:
|
10.21136/CMJ.2017.0063-16 |
| . |
| Date available:
|
2017-06-01T14:31:55Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/146772 |
| . |
| Reference:
|
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| Reference:
|
[2] Andrews, G. E.: Euler's pentagonal number theorem.Math. Mag. 56 (1983), 279-284. Zbl 0523.01011, MR 0720648, 10.2307/2690367 |
| Reference:
|
[3] Berkovich, A., Garvan, F. G.: Some observations on Dyson's new symmetries of partitions.J. Comb. Theory, Ser. A 100 (2002), 61-93. Zbl 1016.05003, MR 1932070, 10.1006/jcta.2002.3281 |
| Reference:
|
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| Reference:
|
[5] Ekhad, S. B., Zeilberger, D.: The number of solutions of $X^2=0$ in triangular matrices over GF($q$).Electron. J. Comb. 3 (1996), Research paper R2, 2 pages printed version in J. Comb. 3 1996 25-26. Zbl 0851.15010, MR 1364064 |
| Reference:
|
[6] Petkovšek, M., Wilf, H. S., Zeilberger, D.: $A=B$.With foreword by Donald E. Knuth, A. K. Peters, Wellesley (1996). Zbl 0848.05002, MR 1379802 |
| Reference:
|
[7] Shanks, D.: A short proof of an identity of Euler.Proc. Am. Math. Soc. 2 (1951), 747-749. Zbl 0044.28403, MR 0043808, 10.1090/S0002-9939-1951-0043808-6 |
| Reference:
|
[8] Warnaar, S. O.: $q$-hypergeometric proofs of polynomial analogues of the triple product identity, Lebesgue's identity and Euler's pentagonal number theorem.Ramanujan J. 8 (2004), 467-474. Zbl 1066.05023, MR 2130521, 10.1007/s11139-005-0275-0 |
| . |
