tailieunhanh - Some new identities in combinatoric

In this paper "Some new identities in combinatoric", the binomial numbers m n are very important in several applications and satisfy several number of identities. The purpose of this paper is to introduce a new combinatorial integer m,n j and obtain some algebraic identities by means of double combinatorial argument. | Journal of Science and Arts Year 13 No. 3 24 pp. 217-230 2013 ORIGINAL PAPER SOME NEW IDENTITIES IN COMBINATORIC DAM VAN NHI 1 TRAN TRUNG TINH1 _ Manuscript Accepted paper Published online 15 . Abstract. In this paper we introduce some new identities in combinatoric. Keywords Equation Identity Combinatoric. 2010 Mathematics Subject Classification 26D05 26D15 51M16. 1. INTRODUCTION n x Proposition . Denote ϕ x α . Then there are the following identities i 1 i n n 1 i ϕ i 1 n . i n i i 0 i n n 1 i n 1 n . n ii i 0 i i n n 1 n 1 1 . n iii i 0 i i n 1 ϕ i i i 4n 1 n ϕ 0 . n iv i 1 2i 1 c 1 n n 2n n i 1 2 n v i 1 i 2i 1 i 2 n . Proposition . There is the following identity 2n n 1 k 2 s2 k 2 n k n k 2n . n 2 s 1 k 0 1 k2 1 Ha Noi National University of Education 136 Xuan Thuy Road Cau Giay Hanoi Vietnam. E-mail damvannhi@. ISSN 1844 9581 Mathematics Section 218 Some new identities in combinatoric Dam Van Nhi Tran Trung Tinh Proposition . For all integer n 1 there is the following identity 2n 1 s 2 k 2 n n k n n k 2n . 2 s 1 k 1 1 k2 Proposition . For all integer n 1 there is the following identity n 2 2n k 1 n k 2 s 2 1 2 k n k 2 n . s 1 1 n 1 k 2 n k 0 2 k 0 1 k2 Proposition . For all integer n 1 we have the following identity 2n 4 2n 1 n k n k n . n n 2 n k 2 r 2 2 1 k 2 n k 0 r 1 1 k2 k 0 2. PROVING SOME NEW IDENTITIES IN COMBINATORIC BY USING THE SYSTEMS OF LINEAR EQUATIONS Example . Assume that α1 α 2 . . . α n and α i j 0 i j 1 2 . . . n . Solve the following system of linear equations x1 x2 xn 1 α 2 α . n α 1 1 1 1 x1 x2 xn . 1 1 α 2 2 α 2 n α2 . x1 x2 xn 1 α 2 α . n α 1 n n n x1 x x p x Proof Consider f x 2 . n 1 where p x is a 1 x 2 x n x n i x i 1 polynomial of degree n. Since f α i 0 therefore p α i 0 i 1 . n . In view of this result we get f x x α1 x α 2 . x α n . x 1 x 2 . x n Mathematics Section Some new identities in combinatoric Dam Van Nhi Tran Trung Tinh 219 From x1 x x 2 . n 1