Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: q-Identities related to overpartitions and divisor functions. | q-Identities related to overpartitions and divisor functions Amy M. Fu Center for Combinatorics LPMC Nankai University Tianjin 300071 . China Email fmu@ Alain Lascoux Nankai University Tianjin 300071 . China Email CNRS IGM Université de Marne-la-Vallée 77454 Marne-la-Vallée Cedex France Abstract. We generalize and prove two conjectures of Corteel and Lovejoy related to overpartitions and divisor functions. Submitted Apr 29 2004 Accepted Jun 16 2005 Published Aug 5 2005 MR Subject Classifications 11B65 41A05 1. Introduction In this paper for any pair positive integers m n we prove the following two identities n X i 1 n if 1 x 1 x qi 1 i 1 - qi m qmi XX -1 i-1 xi - -1 i qi X mij Xi 1 - q jj-jj Q 2 - qi z q n 1 X H -1 i 1 x 1 x qi 1 i X _ 1 i-1 z q i xifji q q n i 0 i 1 - zqi 2-0 q q i using the classical notations z q i 1 - z 1 - zqi 1 and In the next section we shall show that and can be ob n i q q n q q i q q n i. tained from the Newton interpolation in points -1 -q -q2 using the complete symmetric function in the variables q 1 - q q2 1 - q2 . Given X x1 x2 Newton gave the following interpolation formula for any function f x f x f xi f@1 x - X1 f@1@2 x - xi x - X2 - THE ELECTRONIC JOURNAL OF COMBINATORICS 12 2005 R38 1 where dị acting on its left is defined by f . . Xi xi i . - f . . Xi 1 Xi . f Xi . Xi Xi i . . di ------ ----1 ------------------ ----. Xi - Xi 1 Taking f x Xn we have X di di hn-i Xi X2 . Xi i where hk is the complete symmetric function of degree k defined by hk Xi X2 . Xn X -T . -T . . . . -T . J Xii xÌ2 Xik i ii i2 --- ik n Recall the following properties of hk 1. Given an alphabet X the generating function of hk is 1 ề hk X t nxo X In particular Take X fz zq zq2 . Xn fz zq zq2 . zqn i . We have 1 1 k 0 hk z zq zq . k n 0 - tzqz and 1 1 52hk z zq zq . zq- it i- i k 0 Hi Q 1 tzq In consequence of the q-binomial theorem it follows that X a q i ti 1 0 q q i at q i t q i hk z zq zq2 . zk . q
