tailieunhanh - Báo cáo toán học: " Combinatorial and Automated Proofs of Certain Identities"

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: Combinatorial and Automated Proofs of Certain Identities. | Combinatorial and Automated Proofs of Certain Identities Justin Brereton Massachusetts Institute of Technology jbrere@ Maryam Karnib Wayne State University maryamk24@ Alex Quenon Eastern Michigan University altheuser@ Amelia Farid Columbia University amf2153@ Gary Marple Colorado State University-Pueblo marbleandmarple@ Akalu Tefera Grand Valley State University teferaa@ Submitted Apr 29 2011 Accepted Jun 8 2011 Published Jun 18 2011 Mathematics Subject Classification 05A19 Abstract This paper focuses on two binomial identities. The proofs illustrate the power and elegance in enumerative algebraic combinatorial arguments modern machine-assisted techniques of Wilf-Zeilberger and the classical tools of generatingfunctionol-ogy- Key Words and Phrases recurrence equations combinatorial identities Zeilberger Algorithm WZ generatingfunctionology. Dedicated to Doron Zeilberger on the occasion of his sixtieth birthday. 1 Introduction Conjecturing and proving identities of the form A B where A is sum of nice terms such as binomial and B is a closed form or a sum of nice terms is among ancient and attractive mathematical problems. There are several types of proof techniques from THE ELECTRONIC JOURNAL OF COMBINATORICS 18 2 2011 P14 1 various areas of mathematics that can be used to prove such identities. Among these techniques we mention three here enumerative combinatorics generatingfunctionology and the the Wilf-Zeilberger WZ method. Enumerative combinatorics deals with counting the number of certain combinatorial objects. It gives meaning and understanding of such objects and provides an elegant and creative way of verification. Many problems that arise in applications have a relatively simple combinatorial interpretation. For example the binomial coefficient n counts the number of different ways of selecting k objects from a set of n objects. Even though this method gives combinatorial interpretations it is at times