0 10 70 26 v 1 2 6 Ju l 2 00 1 Combinatorial identities for binary necklaces from exact ray - splitting trace formulae

Combinatorial identities for binary necklaces from exact ray-splitting trace formulae

Based on an exact trace formula for a one-dimensional ray-splitting system, we derive novel combinatorial identities for cyclic binary sequences (Pólya necklaces). 02.10.Eb, 02.30.Lt, 05.45.+b Typeset using REVTEX 1

0 40 70 95 v 1 2 6 Ju l 2 00 4 Lectures on Mathematical Cosmology

ar X iv : m at h - ph / 0 60 70 40 v 1 2 0 Ju l 2 00 6 Invariant Form of BK - factorization and its Applications

- th / 0 60 70 56 v 1 2 6 Ju l 2 00 6 SQM 2006 : Theory Summary and Perspectives Steffen

COMBINATORIAL IDENTITIES DERIVING FROM THE n-th POWER OF A 2× 2 MATRIX

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 pow...