TAILIEUCHUNG - Báo cáo toán học: "Ramanujan Type Congruences for a Partition Function"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Department of Mathematic dành cho các bạn yêu thích môn toán học đề tài: Ramanujan Type Congruences for a Partition Function. | Ramanujan Type Congruences for a Partition Function Haijian Zhao and Zheyuan Zhong Center for Combinatorics LPMC-TJKLC Nankai University Tianjin . China zhongzheyuan@ Submitted Nov 8 2010 Accepted Mar 1 2011 Published Mar 11 2011 Mathematics Subject Classifications 05A17 11P83 Abstract We investigate the arithmetic properties of a certain function b n given by ro 52 b n qn q q ro2 q2 q2 ro2. One of our main results is b 9n 7 0 mod 9 . n 0 1 Introduction Recently Chan 5 introduced the function a n which arised from his study of Ramanujan s cubic continued fraction. The function a n is defined by 1 ro q M C q2ỉx n 0a n q Throughout this paper we assume q 1 and we adopt the customary notation ro a q ro n 1 - aqn-1 - n 1 There are many similar properties between a n and the standard partition function p n see 5-9 11 for examples. One of the nice results of a n is the generating function of a 3n 2 obtained by Chan 5 which states that V a 3n 2V q- q3 ro q6 2 ro. 2 00a 3n 2 q 3- q q4 1-1 This identity was also proven by Baruah and Ojah 1 using the 3-dissections for y q -1 and q -1 and by Cao 4 applying the 3-dissection for q q ro q2 q2 ro. We will give another proof based on identities of cubic theta functions in Section 2. THE ELECTRONIC JOURNAL OF COMBINATORICS 18 2011 P58 1 Later Kim 10 studied the following function a n counting the number of overcubic partitions of n y a n qn -q q -q2lq2 n 0a n q q . In this paper we are interested in the function b n defined by q q x n s L3 Our main aim is to present certain arithmetic properties for b n . In Section 3 we will prove the following Ramanujan type congruence modulo 9 that is for any n 0 b 9n 7 0 mod 9 . We also establish two Ramanujan type congruences modulo 5 and 7 by using two classical identities that is for any n 0 b 5n 4 0 mod 5 and b 7n 2 b 7n 3 b 7n 4 b 7n 6 0 mod 7 . 2 Preliminaries In this section we use cubic theta functions to obtain a .

• Báo cáo khoa học
• báo cáo của tạp chí Department of Mathematic
• tài liệu báo cáo nghiên cứu khoa học
• cách trình bày báo cáo
• kiến thức toán học
• báo cáo toán học