Geodesic
Several $q$-series identities from the Euler expansions of $(a;q)_{\infty }$ and $\frac{1}{(a;q)_{\infty }}$
Archivum mathematicum, Tome 45 (2009) no. 1, pp. 47-58 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

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.
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.
Classification : 05A30, 33D15, 33D60
Keywords: exponential operator; operator identity; $q$-series identity 
@article{ARM_2009_45_1_a3,
     author = {Zhang, Zhizheng and Yang, Jizhen},
     title = {Several $q$-series identities from the {Euler} expansions of $(a;q)_{\infty }$ and $\frac{1}{(a;q)_{\infty }}$},
     journal = {Archivum mathematicum},
     pages = {47--58},
     year = {2009},
     volume = {45},
     number = {1},
     mrnumber = {2591660},
     zbl = {1212.05017},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ARM_2009_45_1_a3/}
}
TY  - JOUR
AU  - Zhang, Zhizheng
AU  - Yang, Jizhen
TI  - Several $q$-series identities from the Euler expansions of $(a;q)_{\infty }$ and $\frac{1}{(a;q)_{\infty }}$
JO  - Archivum mathematicum
PY  - 2009
SP  - 47
EP  - 58
VL  - 45
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/ARM_2009_45_1_a3/
LA  - en
ID  - ARM_2009_45_1_a3
ER  - 
%0 Journal Article
%A Zhang, Zhizheng
%A Yang, Jizhen
%T Several $q$-series identities from the Euler expansions of $(a;q)_{\infty }$ and $\frac{1}{(a;q)_{\infty }}$
%J Archivum mathematicum
%D 2009
%P 47-58
%V 45
%N 1
%U http://geodesic.mathdoc.fr/item/ARM_2009_45_1_a3/
%G en
%F ARM_2009_45_1_a3
Zhang, Zhizheng; Yang, Jizhen. Several $q$-series identities from the Euler expansions of $(a;q)_{\infty }$ and $\frac{1}{(a;q)_{\infty }}$. Archivum mathematicum, Tome 45 (2009) no. 1, pp. 47-58. http://geodesic.mathdoc.fr/item/ARM_2009_45_1_a3/

[1] Andrews, G. E.: Ramanujan’s “Lost” Notebook I, Partial $\theta $-function. Adv. Math. 41 (1981), 137–172. | DOI

[2] Andrews, G. E.: Ramanujan: Essays and Surveys. ch. An introduction to Ramanujan's “Lost” Notebook, pp. 165–184, 2001.

[3] Andrews, G. E., Askey, R., Roy, R.: Special Function. Cambridge University Press, Cambridge, 1999.

[4] Chen, W. Y. C., Liu, Z.-G.: Parameter augmentation for basic hypergeometric series, II. J. Combin. Theory Ser. A 80 (1997), 175–195. | DOI | MR | Zbl

[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

[6] Gasper, G. G., Rahman, M.: Basic Hypergeometric Series. Encyclopedia of Mathematics and Its Applications, Second edition, vol. 96, Cambridge University Press, 2004. | MR | Zbl

[7] Liu, Z.-G.: A new proof of the Nassrallah-Rahman integral. Acta Math. Sinica 41 (2) (1998), 405–410. | MR | Zbl

[8] Liu, Z.-G.: Some operator identities and $q$-series transformation formulas. Discrete Math. 265 (2003), 119–139. | DOI | MR | Zbl

[9] Rogers, L. J.: On the expansion of some infinite products. Proc. London Math. Soc. 24 (1893), 337–352.

[10] Rogers, L. J.: Second Memoir on the expansion of certain infinite products. Proc. London Math. Soc. 25 (1894), 318–343.

[11] Rogers, L. J.: Third Memoir on the expansion of certain infinite products. Proc. London Math. Soc. 26 (1896), 15–32.

[12] Roman, S.: More on the umbral calculus, with emphasis on the $q$-umbral calculus. J. Math. Anal. Appl. 107 (1985), 222–254. | DOI | MR | Zbl

[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. | DOI | MR | Zbl

[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. | DOI | MR | Zbl