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 $c_{k,0}$, $c_{k,k}$, $k\in\mathbb{N}$

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

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