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: The Abel-type polynomial identities. | The Abel-type polynomial identities Fengying Huang School of Mathematical Sciences South China Normal University School of Computer Science Guangdong Polytechnic Normal University Guangzhou 510631 . China. E-mail Bolian Liu School of Mathematical Sciences South China Normal University Guangzhou 510631 . China. Corresponding author. E-mail liubl@ Submitted Sep 23 2009 Accepted Dec 29 2009 Published Jan 5 2010 Mathematics Subject Classification 05C30 05C05 Abstract n The Abel identity is x y n n x x iz i-1 y iz n i where x y and z i 0 are real numbers. In this paper we deduce several polynomials expansions referred to as Abel-type identities by using Foata s method and also show some of their applications. 1 Introduction n It is well-known that the binomial identity is x y n y n xiyn-i. In 1826 Abel i 0 deduced an identity which is Ẻ o i 0 k x y n iz i-1 y zz ny 1 where x y and z are real numbers. Then the identity is called Abel identity. When we set z 0 in Eq. 1 it becomes the binomial identity. There are many applications of the Abel identity 1 . And many authors offered different proofs of this identity including the Supported by NNSF of China . THE ELECTRONIC JOURNAL OF COMBINATORICS 17 2010 R10 1 I n I i 1 I I x x iz y nz y iz elegant combinatorial methods by Foata 2 the algebraic method by Lucas 1 and the coding sign method by Francon 1 . In 1996 and presented a computer-generated proof of it 3 . Another well-known version of the classical Abel identity 4 is n x y nz x y n-1 2 i 0 while a generalization of Abel identity expanding a product of multivariate linear forms is Hurwitz identity 1 which is x y x y Z1 Z2 - Zn n-1 y x x Ố1Z1 Ố2 Z2 - enZnY1 2 n 1y y 1Z1 Az 2 y Zn 2 - 1 where the sum is over all 2n possibilities with e1 e2 n choosing 0 or 1 and Si 1 ốj i 1 2 n . All the identities above are dealt with a single summation. In this paper we present three polynomial identities which are .
