Here, we give explicit formulae for solutions of some systems of difference equations, which extend some very particular recent results in the literature and give natural explanations for them, which were omitted in the previous literature.

We show that the system of three difference equationsxn+1=an(1)xn-2/(bn(1)ynzn-1xn-2+cn(1)),yn+1=an(2)yn-2/(bn(2)znxn-1yn-2+cn(2)), andzn+1=an(3)zn-2/(bn(3)xnyn-1zn-2+cn(3)),n∈N0, where all elements of the sequencesan(i),bn(i),cn(i),n∈N0,i∈{1,2,3}, and initial valuesx-j,y-j,z-j,j∈{0,1,2}, are real numbers, can be solved. Explicit formulae for solutions of the system are derived, and some consequences on asymptotic behavior of solutions for the case when coefficients are periodic with period three are deduced.

We obtain a representation of a solution of the Cauchy problem for a linear inhomogeneous differential equation with constant coefficients and pure delay. We use special matrix functions called a delayed matrix sine and a delayed matrix cosine. They have the form of matrix polynomials of degree dependent on the value of delay.

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The purpose of this contribution is to develop a method for construction of solutions of linear discrete systems with constant coefficients and with pure delay. Solutions are expressed with the aid of a special function called the discrete matrix delayed exponential having between every two adjoining knots the form of a polynomial. These polynomials have increasing degrees in the right direction. Such approach results in a possibility to express initial Cauchy problem in the closed form.