Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: Unification of the Quintuple and Septuple Product Identities. | Unification of the Quintuple and Septuple Product Identities Wenchang Chu and Qinglun Yan Department of Applied Mathematics Dalian University of Technology Dalian 116024 P. R. China Submitted Jun 26 2006 Accepted Mar 20 2007 Published Mar 28 2007 Mathematics Subject Classifications 05A30 14K25 Abstract By combining the functional equation method with Jacobi s triple product identity we establish a general equation with five free parameters on the modified Jacobi theta function which can be considered as the common generalization of the quintuple sextuple and septuple product identities. Several known theta function formulae and new identities are consequently proved. 1 Introduction and Notation For two indeterminate q and x with q 1 the q-shifted factorial of infinite order reads as 1 x q i JỊ 1 - xqi 1 - x 1 - qx 1 - qx2 . i 0 Then we define the modified Jacobi theta function by hx q i q q i x q i q x q i. Email addresses and yanqinglun@ THE ELECTRONIC JOURNAL OF COMBINATORICS 14 2007 N7 1 Their product forms are abbreviated respectively as a p 7 q i u q i ft q i 7 q i a ft 7 q i a q i ft q i 7 q i. There are several important theta function identities in mathematical literature. Perhaps the simplest and the most significant one is Jacobi s triple product identity 23 see 18 also 1 x E -1 xn n i For different proofs and applications we refer the reader to the papers by Andrews 2 Chu 8 Ewell 10 Lewis 25 Mordell 27 and Wright 33 . The next one is the celebrated quintuple product identity attributed originally to Watson 32 . However it can also be found in Ramanujan s lost notebook which has been cleared by Berndt 4 P 83 . Later in 1995 Ewell 11 found a sextuple product identity which has both beautiful form and interesting applications to congruences of partition function see Ewell 12 13 14 . Hirschhorn 21 found first in 1983 the septuple product identity which was subsequently rediscovered by Farkas and Kra 15 in 1999. More .
