Solvability of boundary value problems for a class of partial difference equations on the combinatorial domain

By modifying our recent method of half-lines we show how the following boundary value problem for partial difference equations can be solved in closed form: $$\begin{aligned}& d_{n,k}= d_{n-1,k-1}+f(k)d_{n-1,k},\quad 1\le k< n, \\& d_{n,0}=u_{n},\qquad d_{n,n}=v_{n},\quad n \in \mathbb{N}, \end{aligned}$$ where $(u_{n})_{n\in \mathbb{N}}$ and $(v_{n})_{n\in \mathbb{N}}$ are given sequences of complex numbers, and f is a complex-valued function on $\mathbb{N}$ .