# Further Combinatorial Identities Deriving from the n-th Power of a 2 X 2 Matrix

Article

2006

## Abstract

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 X i,j,k,t 2 1+2t−mn+n (−1)nk+i(n+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 ! = mn − 1 − s s ! . 2) The generalized Fibonacci polynomial fm(x, s) can be expressed as fm(x, s) = b(mX−1)/2c k=0 m − k − 1 k ! x m−2k−1 s k . We prove that the following functional equation holds: fmn(x, s) = fm(x, s) × fn ( fm+1(x, s) + sfm−1(x, s), −(−s) m) . 3) If an arithmetical function f is multiplicative and for each prime p there is a complex number g(p) such that f(p n+1) = f(p)f(p n ) − g(p)f(p n−1 ), n ≥ 1, then f is said to be specially multiplicative. We give another derivation of the following formula for a specially multiplicative function f evaluated at a prime power: f(p k ) = b X k/2c j=0 (−1)j k − j j ! f(p) k−2j g(p) j . We also prove various other combinatorial identities.

## Publication Title

Discrete Applied Mathematics

0166-218X

Elsevier

154

8

1301

1308