Note on the binomial partial difference equation

Some formulas for the "general solution" to the binomial partial difference equation $$c_{m,n}=c_{m-1,n}+c_{m-1,n-1},$$ are known in the literature. However, it seems that there is no such a formula on the most natural domain connected to the equation, that is, on the set $D=\big\{(m,n)\in\mathbb{N}^2_0 : 0\le n\le m\big\}.$ By using a connection with the scalar linear first order difference equation we show that the equation on the domain $D\setminus\{(0,0)\}$, can be solved in closed form by presenting a formula for the solution in terms of the "side" values $c_{k,0}$, $c_{k,k}$, $k\in\mathbb{N}$

On an extension of a recurrent relation from combinatorics

The following recurrent relation/partial difference equation $$w_{n,k}=aw_{n-1,k-1}+bw_{n-1,k},$$ where $k,n,a,b\in\mathbb{N}$, appears in a problem in combinatorics. Here we show that an extension of the recurrent relation is solvable on, the, so called, combinatorial domain ${\mathcal C}=\big\{(n,k)\in\mathbb{N}^2_0 : 0\le k\le n\big\}\setminus\{(0,0)\},$ when its coefficients and the boundary values $w_{j,0}$, $w_{j,j}$, $j\in\mathbb{N}$, are complex numbers, by presenting a representation of the general solution to the recurrent relation on the domain in terms of the boundary values. As a special case we obtain a solution to the problem in combinatorics in an elegant way. From the general solution along with an application of the linear first-order difference equation is also obtained the solution of the recurrent relation in the case $w_{j,j}=c\in\mathbb{C}$, $j\in\mathbb{N}$

Product-type system of difference equations of second-order solvable in closed form

This paper presents solutions of the following second-order system of difference equations $$x_{n+1}=\frac{y_n^a}{z_{n-1}^b},\qquad y_{n+1}=\frac{z_n^c}{x_{n-1}^d},\qquad z_{n+1}=\frac{x_n^f}{y_{n-1}^g},\qquad n\in N_0,$$ where $a,b,c,d,f,g\in Z$, and $x_{-i}, y_{-i}, z_{-i}\in C\setminus\{0\},$ $i\in\{0,1\}$, in closed form

On a two-dimensional solvable system of difference equations

Here we solve the following system of difference equations xn+1 = ynyn−2 bxn−1 + ayn−2 , yn+1 = xnxn−2 dyn−1 + cxn−2 , n ∈ N0, where parameters a, b, c, d and initial values x−j , y−j , j = 0, 2, are complex numbers, and give a representation of its general solution in terms of two specially chosen solutions to two homogeneous linear difference equations with constant coefficients associated to the system. As some applications of the representation formula for the general solution we obtain solutions to four very special cases of the system recently presented in the literature and proved by induction, without any theoretical explanation how they can be obtained in a constructive way. Our procedure presented here gives some theoretical explanations not only how the general solutions to the special cases are obtained, but how is obtained general solution to the general system

On a two-dimensional solvable system of difference equations

Here we solve the following system of difference equations $$x_{n+1}=\frac{y_ny_{n-2}}{bx_{n-1}+ay_{n-2}},\quad y_{n+1}=\frac{x_nx_{n-2}}{dy_{n-1}+cx_{n-2}},\quad n\in\mathbb{N}_0,$$ where parameters $a, b, c, d$ and initial values $x_{-j},$ $y_{-j}$, $j=\overline{0,2},$ are complex numbers, and give a representation of its general solution in terms of two specially chosen solutions to two homogeneous linear difference equations with constant coefficients associated to the system. As some applications of the representation formula for the general solution we obtain solutions to four very special cases of the system recently presented in the literature and proved by induction, without any theoretical explanation how they can be obtained in a constructive way. Our procedure presented here gives some theoretical explanations not only how the general solutions to the special cases are obtained, but how is obtained general solution to the general system

On a higher-order system of difference equations

Here we study the following system of difference equations xn = f −1 cnf(xn−2k) ∏k an + bn i=1 g(y) n−(2i−1))f(xn−2i) yn = g −1 γng(yn−2k) ∏k αn + βn i=1 f(x) n−(2i−1))g(yn−2i) n ∈ N0, where f and g are increasing real functions such that f(0) = g(0) = 0, and coefficients an, bn, cn, αn, βn, γn, n ∈ N0, and initial values x−i, y−i, i ∈ {1, 2,..., 2k} are real numbers. We show that the system is solvable in closed form, and study asymptotic behavior of its solutions

New solvable class of product-type systems of difference equations on the complex domain and a new method for proving the solvability

This paper continues the investigation of solvability of product-type systems of difference equations, by studying the following system with two variables: $$z_n=\alpha z_{n-1}^aw_{n-2}^b,\quad w_n=\beta w_{n-3}^cz_{n-2}^d,\quad n\in\mathbb{N}_0,$$ where $a,b,c,d\in\mathbb{Z}$, $\alpha,\beta\in\mathbb{C}\setminus\{0\}$, $w_{-3}, w_{-2}, w_{-1}, z_{-2}, z_{-1}\in\mathbb{C}\setminus\{0\}$. It is shown that there are some important cases such that the system cannot be solved by using our previous methods. Hence, we also present a method different from the previous ones by which the solvability of the system is shown also in the cases

On a practically solvable product-type system of difference equations of second order

The problem of solvability of the following second order system of difference equations z(n+1) = alpha Z(n)(a)w(n)(b), w(n+1) = beta w(n)(c)z(n-1)(d), n is an element of N-0, where a, b, c, d is an element of Z, alpha, beta is an element of C \ {0}, z(-1), z(0), w(0) is an element of C \ {0}, is studied in detail