On a solvable system of difference equations of kth order

Abstract

It is shown that the next system of k difference equationsxn(j)=an(j)xn- k(j)bn(j)∏i=1kxn-i(σ(j+i-1))+cn(j),n∈N0,j=1,k,where an(j),bn(j),cn(j),n∈N0,j=1,k, and initial values x-i(j),i,j∈{1,...,k}, are real numbers, and where σ:N→{1,...,k} is defined by σ(km+j)=j+1,j=1,k-1,σ(km+k)=1, m∈N0, can be solved in closed form in an elegant way. Some applications of obtained formulas, for the case when the sequences an(j),bn(j) and cn(j) are k-periodic are also given.