On a solvable system of difference equations of kth order

摘要

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.