Báo cáo toán học: "The Abel-type polynomial 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: 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 .

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