TY - JOUR
T1 - An infinite family of engel expansions of Rogers-Ramanujan type
AU - Andrews, George E.
AU - Knopfmacher, Arnold
AU - Paule, Peter
N1 - Funding Information:
DEDICATED TO THE MEMORY OF JOHN KNOPFMACHER 1937–1999, THE INVENTOR OF ENGEL EXPANSIONS FOR q-SERIES The extended Engel expansion is an algorithm that leads to unique series expansions of q-series. Various examples related to classical partition theorems, including the Rogers–Ramanujan identities, have been given recently. The object of this paper is to show that the new and elegant Rogers–Ramanujan generalization found by Garrett, Ismail, and Stanton also ts into this framework. This not only reveals the existence of an in nite, parameterized family of extended Engel expansions, but 1Partially supported by National Science Foundation Grant DMS-9870060. 2Partially supported by SFB Grant F1305 of the Austrian FWF and by the Applicable Analysis and Number Theory of the University of Witwatersrand.
PY - 2000/7
Y1 - 2000/7
N2 - The extended Engel expansion is an algorithm that leads to unique series expansions of q-series. Various examples related to classical partition theorems, including the Rogers-Ramanujan identities, have been given recently. The object of this paper is to show that the new and elegant Rogers-Ramanujan generalization found by Garrett, Ismail, and Stanton also fits into this framework. This not only reveals the existence of an infinite, parameterized family of extended Engel expansions, but also provides an alternative proof of the Garrett, Ismail, and Stanton result. A finite version of it, which finds an elementary proof, is derived as a by-product of the Engel approach.
AB - The extended Engel expansion is an algorithm that leads to unique series expansions of q-series. Various examples related to classical partition theorems, including the Rogers-Ramanujan identities, have been given recently. The object of this paper is to show that the new and elegant Rogers-Ramanujan generalization found by Garrett, Ismail, and Stanton also fits into this framework. This not only reveals the existence of an infinite, parameterized family of extended Engel expansions, but also provides an alternative proof of the Garrett, Ismail, and Stanton result. A finite version of it, which finds an elementary proof, is derived as a by-product of the Engel approach.
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U2 - 10.1006/aama.2000.0686
DO - 10.1006/aama.2000.0686
M3 - Article
AN - SCOPUS:0034216096
SN - 0196-8858
VL - 25
SP - 2
EP - 11
JO - Advances in Applied Mathematics
JF - Advances in Applied Mathematics
IS - 1
ER -