COMBINATORIAL IDENTITIES DERIVING FROM THE n-th POWER OF A 2× 2 MATRIX
نویسنده
چکیده
In this paper we give a new formula for the n-th power of a 2× 2 matrix. More precisely, we prove the following: Let A = ( a b c d ) be an arbitrary 2×2 matrix, T = a+ d its trace, D = ad− bc its determinant and define yn : = bn/2c ∑ i=0 ( n− i i ) T n−2i(−D)i. Then, for n ≥ 1, A = ( yn − d yn−1 b yn−1 c yn−1 yn − a yn−1 ) . We use this formula together with an existing formula for the n-th power of a matrix, various matrix identities, formulae for the n-th power of particular matrices, etc, to derive various combinatorial identities.
منابع مشابه
Further combinatorial identities deriving from the nth power of a 2×2 matrix
In this paper we use a formula for the n-th power of a 2×2 matrix A (in terms of the entries in A) to derive various combinatorial identities. Three examples of our results follow. 1) We show that if m and n are positive integers and s ∈ {0, 1, 2, . . . , b(mn− 1)/2c}, then ∑ i,j,k,t 2 (−1) 1 + δ(m−1)/2, i+k ( m− 1− i i )( m− 1− 2i k ) × ( n(m− 1− 2(i+ k)) 2j )( j t− n(i+ k) )( n− 1− s+ t s− t )
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تاریخ انتشار 2004