Further combinatorial identities deriving from the n th power of a 2×22×2 matrix

In this paper we use a formula for the n  th power of a 2×22×2 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,...,s∈{0,1,2,...,⌊(mn-1)/2⌋}⌊(mn-1)/2⌋}, then∑i,j,k,t21+2t-mn+n(-1)nk+i(n+1)1+δ(m-1)/2,i+km-1-iim-1-2ikn(m-1-2(i+k))2jjt-n(i+k)n-1-s+ts-t=mn-1-ss.(2) The generalized Fibonacci polynomial fm(x,s)fm(x,s) can be expressed asfm(x,s)=∑k=0⌊(m-1)/2⌋m-k-1kxm-2k-1sk.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).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)g(p) such thatf(pn+1)=f(p)f(pn)-g(p)f(pn-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(pk)=∑j=0⌊k/2⌋(-1)jk-jjf(p)k-2jg(p)j.We also prove various other combinatorial identities.