| Title:
|
Several $q$-series identities from the Euler expansions of $(a;q)_{\infty }$ and $\frac{1}{(a;q)_{\infty }}$ (English) |
| Author:
|
Zhang, Zhizheng |
| Author:
|
Yang, Jizhen |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
45 |
| Issue:
|
1 |
| Year:
|
2009 |
| Pages:
|
47-58 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
In this paper, we first give several operator identities which extend the results of Chen and Liu, then make use of them to two $q$-series identities obtained by the Euler expansions of $(a;q)_{\infty }$ and $\frac{1}{(a;q)_{\infty }}$. Several $q$-series identities are obtained involving a $q$-series identity in Ramanujan’s Lost Notebook. (English) |
| Keyword:
|
exponential operator |
| Keyword:
|
operator identity |
| Keyword:
|
$q$-series identity |
| MSC:
|
05A30 |
| MSC:
|
33D15 |
| MSC:
|
33D60 |
| idZBL:
|
Zbl 1212.05017 |
| idMR:
|
MR2591660 |
| . |
| Date available:
|
2009-06-25T13:47:17Z |
| Last updated:
|
2013-09-19 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/128289 |
| . |
| Reference:
|
[1] Andrews, G. E.: Ramanujan’s “Lost” Notebook I, Partial $\theta $-function.Adv. Math. 41 (1981), 137–172. 10.1016/0001-8708(81)90013-X |
| Reference:
|
[2] Andrews, G. E.: Ramanujan: Essays and Surveys.ch. An introduction to Ramanujan's “Lost” Notebook, pp. 165–184, 2001. |
| Reference:
|
[3] Andrews, G. E., Askey, R., Roy, R.: Special Function.Cambridge University Press, Cambridge, 1999. |
| Reference:
|
[4] Chen, W. Y. C., Liu, Z.-G.: Parameter augmentation for basic hypergeometric series, II.J. Combin. Theory Ser. A 80 (1997), 175–195. Zbl 0901.33009, MR 1485133, 10.1006/jcta.1997.2801 |
| Reference:
|
[5] Chen, W. Y. C., Liu, Z.-G.: Mathematical essays in honor of Gian-Carlo Rota.ch. Parameter augmentation for basic hypergeometric series, I, pp. 111–129, Birkhäuser, Basel, 1998. MR 1627355 |
| Reference:
|
[6] Gasper, G. G., Rahman, M.: Basic Hypergeometric Series.Encyclopedia of Mathematics and Its Applications, Second edition, vol. 96, Cambridge University Press, 2004. Zbl 1129.33005, MR 2128719 |
| Reference:
|
[7] Liu, Z.-G.: A new proof of the Nassrallah-Rahman integral.Acta Math. Sinica 41 (2) (1998), 405–410. Zbl 1010.33009, MR 1656510 |
| Reference:
|
[8] Liu, Z.-G.: Some operator identities and $q$-series transformation formulas.Discrete Math. 265 (2003), 119–139. Zbl 1021.05010, MR 1969370, 10.1016/S0012-365X(02)00626-X |
| Reference:
|
[9] Rogers, L. J.: On the expansion of some infinite products.Proc. London Math. Soc. 24 (1893), 337–352. |
| Reference:
|
[10] Rogers, L. J.: Second Memoir on the expansion of certain infinite products.Proc. London Math. Soc. 25 (1894), 318–343. |
| Reference:
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[11] Rogers, L. J.: Third Memoir on the expansion of certain infinite products.Proc. London Math. Soc. 26 (1896), 15–32. |
| Reference:
|
[12] Roman, S.: More on the umbral calculus, with emphasis on the $q$-umbral calculus.J. Math. Anal. Appl. 107 (1985), 222–254. Zbl 0654.05004, MR 0786026, 10.1016/0022-247X(85)90367-1 |
| Reference:
|
[13] Zhang, Z. Z., Liu, M. X.: Applications of operator identities to the multiple $q$-binomial theorem and $q$-Gauss summation theorem.Discrete Math. 306 (2006), 1424–1437. Zbl 1095.05002, MR 2237725, 10.1016/j.disc.2006.01.025 |
| Reference:
|
[14] Zhang, Z. Z., Wang, J.: Two operator identities and their applications to terminating basic hypergeometric series and $q$-integrals.J. Math. Anal. Appl. 312 (2) (2005), 653–665. Zbl 1081.33032, MR 2179103, 10.1016/j.jmaa.2005.03.064 |
| . |
