\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2017 (2017), No. 17, pp. 1--10.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2017 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2017/17\hfil Partial difference equations]
{Solvability of boundary-value problems for a linear partial difference equation}

\author[S. Stevi\'c \hfil EJDE-2017/17\hfilneg]
{Stevo Stevi\'c}

\address{Stevo Stevi\'c \newline
 Mathematical Institute of the Serbian Academy of Sciences,
 Knez Mihailova 36/III, 11000 Beograd, Serbia \newline
Operator Theory and Applications Research Group,
Department of Mathematics, King Abdulaziz University,
P.O. Box 80203, Jeddah 21589, Saudi Arabia}
 \email{sstevic@ptt.rs}

\dedicatory{Communicated by Vicentiu Radulescu}

\thanks{Submitted October 23, 2016. Published January 14, 2017.}
\subjclass[2010]{39A14, 39A06}
\keywords{Partial difference equation; solvable difference equation;
\hfill\break\indent method of half-lines; combinatorial domain}

\begin{abstract}
 In this article we consider the  two-dimensional boundary-value problem
 \begin{gather*}
 d_{m,n}=d_{m-1,n}+f_nd_{m-1,n-1},\quad 1\le n<m,\\
 d_{m,0}=a_m,\quad d_{m,m}=b_m,\quad m\in\mathbb{N},
 \end{gather*}
 where $a_m$, $b_m$, $m\in\mathbb{N}$ and $f_n$, $n\in\mathbb{N}$,
 are complex sequences. Employing recently introduced method of half-lines,
 it is shown that the boundary-value problem is solvable,
 by finding an explicit formula for its solution on the domain,
 the, so called, combinatorial domain. The problem is solved
 for each complex sequence $f_n$, $n\in\mathbb{N}$, that is, even
 if some of its members are equal to zero.
 The main result here extends a recent result in the topic.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{remark}[theorem]{Remark}
\newtheorem{corollary}[theorem]{Corollary}
\allowdisplaybreaks

\section{Introduction}

Let $\mathbb{N}$ be the set of all natural numbers,
$\mathbb{N}_0=\mathbb{N}\cup\{0\}$ and $\mathbb{Z}$ the set of all integers.
 If $k, l\in\mathbb{Z}$ and $k\le l$, then by $\overline{k,l}$, we will denote
the set of all integers $j$, such that $k\le j\le l$.

It is well-known that the binomial coefficients $C^m_n$, where
$n,m\in\mathbb{N}_0$ are such that $0\le n\le m$, satisfy the following relations:
\begin{equation}
C^m_0=C^{m-1}_0,\quad C^m_m=C^{m-1}_{m-1},\label{a0}
\end{equation}
for $m\ge 2$ and
\begin{equation}
 C^m_n=C^{m-1}_n+C^{m-1}_{n-1},\label{b}
\end{equation}
for every $m,n\in\mathbb{N}$ such that $1\le n\le m-1$ and $m\ge 2$,
(see for example the books \cite{k, mk1, mk, ri, ya},
 where these and many other relations connected to the  binomial
coefficients can be found).
In other words, the three relations in
 \eqref{a0} and \eqref{b} mean that the sequence $C^m_n$
(with two independent variables $m$ and $n$) is the solution to the
following boundary-value problem for partial difference equations
\begin{equation}
\begin{gathered}
c_{m,n}=c_{m-1,n}+c_{m-1,n-1},\quad 1\le n<m,\\
c_{m,0}=1,\quad c_{m,m}=1,\quad m\in\mathbb{N}.
\end{gathered} \label{bpdce}
\end{equation}

This fact along with the existence of a closed form formula for the sequence
$C^m_n$, that is, the formula
$$
C^m_n=\frac{m!}{n!(m-n)!},\quad 0\le n\le m,
$$
where by definition is regarded that $0!=1$, has suggested us that
the values of the quantities $C_0^m$ and $C_m^m$, $m\in\mathbb{N}$,
that is, the values of $c_{m,0}$ and $c_{m,m}$, $m\in\mathbb{N}$,
essentially do not influence on the \emph{solvability} of the following
 boundary-value problem for partial difference equations
\begin{equation}
\begin{gathered}
c_{m,n}=c_{m-1,n}+c_{m-1,n-1},\quad 1\le n<m,\\
c_{m,0}=u_m,\quad c_{m,m}=v_m,\quad m\in\mathbb{N},
\end{gathered}\label{bpdce1}
\end{equation}
and, moreover, that these quantities need not be only integers or real numbers,
but can be even complex numbers, that is, that given sequences
$(u_m)_{m\in\mathbb{N}}$ and $(v_m)_{m\in\mathbb{N}}$ can be complex.

It is known that the partial difference equation appearing in
\eqref{bpdce} and \eqref{bpdce1}, which we call the
\emph{binomial partial difference equation}, is ``solvable"
(see for example \cite{ll} for a method for solving the equation).
However, the notion of solvability of partial difference equations highly
depends on the domain in which the equation is treated
(see for example \cite{ch}), which is why we have written the word \emph{solvable}
under the signs of quotations.

The following formula for ``general solution" to the binomial partial
difference equation
$$
c_{m,n}=\sum_{j=0}^mC^m_jc_{0,n-j},
$$
can be found in the literature (see for example \cite{ll}).
 However, since the last formula depends on the values of the quantities
$c_{0,l}$, $l\in\mathbb{Z}$, only, which lie on the same line,
the formula can be regarded as a general solution to the equation only
on a half plane.

All above mentioned have motivated us to show the solvability of boundary-value
problems for the binomial partial difference equation on its natural domain,
that is, on the domain
$$
\mathcal{C}=\{(m,n): 0\le n\le m,\, m,n\in\mathbb{N}_0\}\setminus\{(0,0)\},
$$
which we call, the \emph{combinatorial domain}.
The solvability of the boundary-value problem for the equation on the domain
was shown in our paper \cite{ejqtde-pdce}, where we devised a method,
which we call, the \emph{method of half-lines}.
There are several ideas behind the method. One of the main ideas is to
slice  the combinatorial domain on half-lines and consider a partial
difference equation on them, but as one-dimensional (scalar)
difference equations. The other idea is to try to solve the scalar equations,
but if they are not solvable then we will write them in a form that
looks like as solvable ones, then ``solve" them and by using posed boundary
conditions to get a solution on the half-lines. Finally, based on
such obtained formulas on the half-lines, it should be concluded the form
of the general solution for the boundary-value problem on the domain.

Studying difference equations of various types is an area of considerable
interest, especially in the last few last decades
(see for example \cite{ra}-\cite{mm}, \cite{mk}-\cite{ade1}
and the references therein).

Some recent investigations of solvable difference equations show that
the nonhomogeneous linear first order difference equation
(with variable coefficients), that is, the  equation
\begin{equation}
x_n=a_nx_{n-1}+b_n,\quad n\in\mathbb{N},\label{lfo}
\end{equation}
where coefficients $a_n$ and $b_n$, $n\in\mathbb{N},$ and initial value $x_0$
are real or complex numbers, plays a very important role in the solvability
of many classes of differences equations.
The method of transformation has been used successfully and developed recently
by several authors in numerous papers on difference equations
(see for example \cite{pst2, ana1, ejde1}),
as well as on papers on systems of difference equations
(see for example \cite{amc218-sde, amc219-sstho, amc219-ssktho, jdea205, ade1},
see also numerous related references therein; a considerable interest
to concrete symmetric-type difference equations started after
the publication of papers \cite{ps1}-\cite{ps3} by Papaschinopoulos and Schinas).
 The most important thing connected to equation \eqref{lfo} is that it is
solvable in closed form. For some methods for solving the equation
see, for example, \cite{ra, lb, mk}. For periodic solutions to the equation,
see \cite{ap}. For some classical results on solvability see, for example,
\cite{ra, lb1, lb, j, k, ll, mk}. Recall, that the general solution to the
difference equation is
\begin{equation}
x_n=x_0\prod_{j=1}^na_j+\sum_{i=1}^nb_i\prod_{j=i+1}^na_j,\label{lfo1}
\end{equation}
for $n\in \mathbb{N}_0$.

One of the methods for solving the equation corresponds to the method of
integrating factors for solving the linear first-order differential equation.
 It is interesting to note that the form of general solution to equation
\eqref{lfo} given in \eqref{lfo1} does not exclude the case when some of
 $a_n$-s are equal to zero, which is important here, since we will not have
any restrictions in dealing with the main partial difference equation
in this paper. Namely, if
\begin{equation}
a_{n_0}=0\quad\text{for some } n_0\in\mathbb{N},\label{b0}
\end{equation}
then from \eqref{lfo} with $n=n_0$, we have
$x_{n_0}=b_{n_0}$,
and consequently
$$
x_{n_0+1}=a_{n_0+1}b_{n_0}+b_{n_0+1},
$$
which, on the first site, looks quite different from formula \eqref{lfo1} ($x_{n_0}$, that is, $b_{n_0}$ here looks like a new (shifted) initial value). However, by using formula \eqref{lfo1}, it follows that
\begin{align*}
x_{n_0+1}
&=x_0\prod_{j=1}^{n_0+1}a_j+\sum_{i=1}^{n_0+1}b_i\prod_{j=i+1}^{n_0+1}a_j,\\
&=\sum_{i=n_0}^{n_0+1}b_i\prod_{j=i+1}^{n_0+1}a_j\\
&=b_{n_0}a_{n_0+1}+b_{n_0+1},
\end{align*}
since
$$
\prod_{j=i+1}^{n_0+1}a_j=0,\quad\text{for } i=\overline{1,n_0-1},
$$
by assumption \eqref{b0}.

So, whether or not some of $a_n$-s are equal to zero, formula \eqref{lfo1} holds,
unlike the following formula
$$
x_n=\prod_{j=1}^na_j\Big(x_0+\sum_{i=1}^n\frac{b_i}{\prod_{j=1}^ia_j}\Big),\quad
n\in\mathbb{N}_0,
$$
which holds only if
$$
a_n\ne 0,\quad\text{for every } n\in\mathbb{N}_0.
$$
Assume that $(n_l)_{l\in I}\subseteq \mathbb{N}$,
$\operatorname{card}(I)\le \aleph_0$, is the set of all indices such that
$a_{n_l}=0$, $l\in I,$ and $n_l<n_{l+1}$, for every $l\in I$.
 Then compact formula \eqref{lfo1} can be written as follows
$$
x_{n_l+k}=b_{n_l}\prod_{j=1}^ka_{n_l+j}
+\sum_{j=1}^kb_{n_l+j}\prod_{i=j+1}^ka_{n_l+i},
$$
for every $l\in I$, and $k=\overline{0,n_{l+1}-n_l-1}$.

Our aim here is to show, by using the method of half-lines, that there is
a class of partial difference equations, which includes the binomial
partial difference equation, which is also solvable on the combinatorial domain,
extending the main results in \cite{ejqtde-pdce}.
A problem of this type has been recently treated in \cite{ade-2016pdce}.
However, the present paper can be regarded as the first one which applies
the method of half-lines in a full generality, in the sense that is
applied the general form of the solution to  the linear difference
equation in \eqref{lfo1}, unlike the ones in our previous papers in
the topic (\cite{ejqtde-pdce, ade-2016pdce}), where essentially some
sorts of summing by using the telescoping method is employed.
Our results can be regarded also as a continuation of investigation
of the problem of solvability of difference equations, including
partial difference equations
(\cite{amc218-sde}, \cite{amc219-sstho}-\cite{ade1}).
 For  some classical results on the solvability of partial difference equations
see, for example, \cite{ch, j, ll}, while some results up to 2003,
can be found in monograph \cite{ch} (see also the related references therein,
such as \cite{chu}). Some other types of partial difference equations can
be found, e.g., in \cite{am, rv, rr1}. Some partial difference equations can
be find also in \cite{ra, k, mk1, mk, ri, ya}, but usually in passing,
and they are presented and treated more as some exotic recurrent relations.

\section{Main results}

This section proves the main result in this article and gives some interesting
consequences. Namely, we show that the  boundary-value problem
for partial difference equations
\begin{gather}
d_{m,n}=d_{m-1,n}+f_nd_{m-1,n-1},\quad 1\le n<m,\label{a3}\\
d_{m,0}=a_m,\quad d_{m,m}=b_m,\quad m\in\mathbb{N},\label{a3a}
\end{gather}
where $a_m,$ $b_m$ and $f_m$, $m\in\mathbb{N}$, are complex sequences,
is solvable.

To do this we present the first several steps of the method of half-lines,
for the benefit of the reader and since it is not so immediately clear
how to guess the formula for the boundary-value problem \eqref{a3}-\eqref{a3a},
to avoid presenting a relatively complicated formula on the spot.

If $m=n+1$, then equation \eqref{a3} is
\begin{equation}
d_{n+1,n}=d_{n,n}+f_nd_{n,n-1},\label{e2}
\end{equation}
for $n\in\mathbb{N}$.

If we use the change of variables $x_n=d_{n+1,n}$, then
\eqref{e2} can be regarded as an equation of type \eqref{lfo}
with
$$
a_n=f_n\quad\text{and}\quad b_n=d_{n,n},\quad n\in\mathbb{N}.
$$
If we solve it by using formula \eqref{lfo1}, we obtain
\begin{equation}
d_{n+1,n}=\sum_{j=1}^n
d_{j,j}\prod_{i=j+1}^nf_i+d_{1,0}\prod_{i=1}^nf_i,\label{a7}
\end{equation}
for $n\in\mathbb{N}_0$.

If $m=n+2$, then \eqref{a3} is
\begin{equation}
d_{n+2,n}=d_{n+1,n}+f_nd_{n+1,n-1},\label{a8}
\end{equation}
for $n\in\mathbb{N}$.
Using the change of variables $x_n=d_{n+2,n}$, equation \eqref{a8}
can be regarded as an equation of type \eqref{lfo} with
$$
a_n=f_n\quad\text{and}\quad b_n=d_{n+1,n},\quad n\in\mathbb{N}.
$$
By using formula \eqref{lfo1}, we obtain
\begin{equation}
d_{n+2,n}=\sum_{j=1}^n
d_{j+1,j}\prod_{i=j+1}^nf_i+d_{2,0}\prod_{i=1}^nf_i,\label{a9}
\end{equation}
for $n\in\mathbb{N}_0$.
Using \eqref{a7} with $n=j$ in \eqref{a9}, we obtain
\begin{equation}
d_{n+2,n}=\sum_{j=1}^n \Big(\sum_{l=1}^j
d_{l,l}\prod_{s=l+1}^jf_s+d_{1,0}
\prod_{i=1}^jf_i\Big)\prod_{i=j+1}^nf_i+d_{2,0}\prod_{i=1}^nf_i,\label{a10}
\end{equation}
for $n\in\mathbb{N}_0$.
We have
\begin{equation}
\sum_{j=1}^n\prod_{i=1}^jf_i\prod_{i=j+1}^nf_i
=\sum_{j=1}^n\prod_{i=1}^nf_i=n\prod_{i=1}^nf_i,\label{b1}
\end{equation}
and
\begin{equation}
\begin{aligned}
\sum_{j=1}^n\prod_{i=j+1}^nf_i\sum_{l=1}^j
d_{l,l}\prod_{s=l+1}^jf_s
&=\sum_{l=1}^nd_{l,l}\sum_{j=l}^n\prod_{s=l+1}^jf_s\prod_{i=j+1}^nf_i\\
&=\sum_{l=1}^nd_{l,l}\sum_{j=l}^n\prod_{s=l+1}^nf_s\\
&=\sum_{l=1}^nd_{l,l}(n-l+1)\prod_{s=l+1}^nf_s,
\end{aligned}\label{b2}
\end{equation}
for $n\in\mathbb{N}_0$.
Employing \eqref{b1} and \eqref{b2} in \eqref{a10}, we obtain
\begin{equation}
d_{n+2,n}=\sum_{l=1}^nd_{l,l}(n-l+1)\prod_{s=l+1}^nf_s+d_{1,0}n
\prod_{i=1}^nf_i+d_{2,0}\prod_{i=1}^nf_i,\label{b3}
\end{equation}
for $n\in\mathbb{N}_0$.

If $m=n+3$, then \eqref{a3} is
\begin{equation}
d_{n+3,n}=d_{n+2,n}+f_nd_{n+2,n-1},\label{a11}
\end{equation}
for $n\in\mathbb{N}$.
Using the change of variables $x_n=d_{n+3,n}$, equation
\eqref{a11} can be regarded as an equation of type \eqref{lfo}
with
$$
a_n=f_n\quad\text{and}\quad b_n=d_{n+2,n},\quad n\in\mathbb{N}.
$$
By using formula \eqref{lfo1}, we obtain
\begin{equation}
d_{n+3,n}=\sum_{j=1}^n
d_{j+2,j}\prod_{i=j+1}^nf_i+d_{3,0}\prod_{i=1}^nf_i,\label{a12}
\end{equation}
for $n\in\mathbb{N}_0$.
Using \eqref{b3} with $n=j$ in \eqref{a12}, we obtain
\begin{equation}
\begin{aligned}
d_{n+3,n}
&=\sum_{j=1}^n\Big(\sum_{l=1}^jd_{l,l}(j-l+1)
\prod_{s=l+1}^jf_s+d_{1,0}j\prod_{i=1}^jf_i+d_{2,0}\prod_{i=1}^jf_i\Big) \\
&\quad\times \prod_{i=j+1}^nf_i
+d_{3,0}\prod_{i=1}^nf_i,
\end{aligned} \label{a14}
\end{equation}
for $n\in\mathbb{N}_0$.
We have
\begin{equation}
\sum_{j=1}^nj\prod_{i=1}^jf_i\prod_{i=j+1}^nf_i
=\sum_{j=1}^nj\prod_{i=1}^nf_i=\frac{n(n+1)}2\prod_{i=1}^nf_i,\label{b5}
\end{equation}
and
\begin{equation}
\begin{aligned}
&\sum_{j=1}^n\prod_{i=j+1}^nf_i\sum_{l=1}^jd_{l,l}(j-l+1)\prod_{s=l+1}^jf_s \\
&=\sum_{l=1}^nd_{l,l}\sum_{j=l}^n(j-l+1)\prod_{i=j+1}^nf_i\prod_{s=l+1}^jf_s\\
&=\sum_{l=1}^nd_{l,l}\prod_{s=l+1}^nf_s\sum_{j=l}^n(j-l+1)\\
&=\sum_{l=1}^nd_{l,l}\prod_{s=l+1}^nf_s\sum_{s=1}^{n-l+1}s\\
&=\sum_{l=1}^nd_{l,l}\prod_{s=l+1}^nf_s\frac{(n-l+1)(n-l+2)}2,
\end{aligned} \label{b6}
\end{equation}
for $n\in\mathbb{N}_0$.
By using \eqref{b1}, \eqref{b5} and \eqref{b6} in \eqref{a14}, we
obtain
\begin{equation}
d_{n+3,n}=\sum_{l=1}^nd_{l,l}C^{n-l+2}_2
\prod_{s=l+1}^nf_s+\big(d_{1,0}C^{n+1}_2+d_{2,0}C^n_1+d_{3,0}\big)
\prod_{i=1}^nf_i,\label{b7}
\end{equation}
for $n\in\mathbb{N}_0$.

Based on the formulas \eqref{a7}, \eqref{b3} and \eqref{b7}, we
may assume that
\begin{equation}
\begin{aligned}
d_{n+k,n}
&=\sum_{j=1}^nC^{n-j+k-1}_{k-1}d_{j,j}\prod_{s=j+1}^nf_s\\
&\quad +\big(C^{n+k-2}_{k-1}d_{1,0}+C^{n+k-3}_{k-2}d_{2,0}+\cdots
+C^{n-1}_0d_{k,0}\big)\prod_{i=1}^nf_i,
\end{aligned}\label{a15}
\end{equation}
for every $k,n\in\mathbb{N}$.
Formulas given in \eqref{a7}, \eqref{b3} and \eqref{b7} show that
the equality in \eqref{a15} holds for $k=\overline{1,3}$.

If $m=n+k+1$, then we have
\begin{equation}
d_{n+k+1,n}=d_{n+k,n}+f_nd_{n+k,n-1},\label{a16}
\end{equation}
for $n\in\mathbb{N}$.
Using the change of variables
$x_n=d_{n+k+1,n}$, equation \eqref{a16} can be regarded as an
equation of type \eqref{lfo} with
$$
a_n=f_n\quad\text{and}\quad b_n=d_{n+k,n},\quad n\in\mathbb{N}.
$$
By using formula \eqref{lfo1}, we obtain
\begin{equation}
d_{n+k+1,n}=\sum_{j=1}^n
d_{j+k,j}\prod_{s=j+1}^nf_s+d_{k+1,0}\prod_{s=1}^nf_s,\label{a17}
\end{equation}
for $n\in\mathbb{N}_0$.
Employing \eqref{a15} with $n=j$ in \eqref{a17}, and by some simple
calculations, it follows that
\begin{align}
d_{n+k+1,n}
&=\sum_{j=1}^n\prod_{s=j+1}^nf_s\sum_{i=1}^jC^{j-i+k-1}_{k-1}d_{i,i}
 \prod_{s=i+1}^jf_s \nonumber \\
&\quad +\sum_{j=1}^n\big(C^{j+k-2}_{k-1}d_{1,0}+C^{j+k-3}_{k-2}d_{2,0}+\cdots 
 \nonumber\\
&\quad +C^{j-1}_0d_{k,0}\big)\prod_{s=1}^nf_s
 +d_{k+1,0}\prod_{s=1}^nf_s \nonumber\\
&=\sum_{i=1}^nd_{i,i}\sum_{j=i}^nC^{j-i+k-1}_{k-1}\prod_{s=i+1}^nf_s \nonumber\\
&\quad +\Big(\sum_{r=2}^{k+1}d_{r-1,0}\sum_{j=1}^nC^{j+k-r}_{k-r+1}
 +d_{k+1,0}\Big)\prod_{s=1}^nf_s,  \label{a18}
\end{align}
for $n\in\mathbb{N}_0$.
By using the relation \eqref{b}, we have
\begin{equation}
\begin{aligned}
\sum_{j=1}^nC^{j+k-r}_{k-r+1}
&=\sum_{j=1}^n\Big(C^{j+k-r+1}_{k-r+2}-C^{j+k-r}_{k-r+2}\Big)\\
&=C^{n+k-r+1}_{k-r+2}-C^{k-r+1}_{k-r+2}
=C^{n+k-r+1}_{k-r+2},
\end{aligned} \label{f2}
\end{equation}
 for every $2\le r\le k+1$, and
\begin{equation}
\sum_{j=i}^nC^{j-i+k-1}_{k-1}
=\sum_{j=i}^n\Big(C^{j-i+k}_k-C^{j-i+k-1}_k\Big)
=C^{n-i+k}_k-C^{k-1}_k
=C^{n-i+k}_k, \label{f3}
\end{equation}
for every $1\le i\le n$.
Using \eqref{f2} and \eqref{f3} into \eqref{a18}, it follows that
\begin{equation}
\begin{aligned}
d_{n+k+1,n}
&=\sum_{j=1}^nC^{n-j+k}_kd_{j,j}\prod_{s=j+1}^nf_s\\
&\quad +\big(C^{n+k-1}_kd_{1,0}+C^{n+k-2}_{k-1}d_{2,0}+\cdots+C^{n-1}_0d_{k+1,0}
\big)\prod_{s=1}^nf_s,
\end{aligned}
\end{equation}
from which along with the method of induction it follows that
formula \eqref{a15} holds for every $k,n\in\mathbb{N}$.

The above described process leads us into a position to formulate and prove
 the main result in this note.

\begin{theorem} \label{thm1}
If $(a_k)_{k\in\mathbb{N}}$, $(b_k)_{k\in\mathbb{N}}$, are given sequences
of complex numbers. Then the solution to partial difference equation \eqref{a3} on
domain $\mathcal{C}$, with the boundary conditions
\begin{equation}
 d_{k,0}=a_k,\quad d_{k,k}=b_k,\quad k\in\mathbb{N},\label{a19}
\end{equation}
is given by
\begin{equation}
d_{m,n}=\sum_{j=1}^nC^{m-j-1}_{m-n-1}b_j
\prod_{s=j+1}^nf_s+\sum_{j=1}^{m-n}C^{m-1-j}_{m-n-j}a_j\prod_{s=1}^nf_s.\label{a20}
\end{equation}
\end{theorem}

\begin{proof}
If we put $m=n+k$ in \eqref{a15} and use the
conditions in \eqref{a19}, we obtain formula
\eqref{a20}.
\end{proof}

\begin{remark} \label{rmk1} \rm
 Note that the hypothesis for the solution to boundary-value problem
\eqref{a3}-\eqref{a3a} has become clearer after three steps, that is,
 after finding the ``solutions" to the corresponding first-order linear
difference equations on the half-lines $m=n+j$, $j=\overline{1,3}$,
$n\in\mathbb{N}_0$.
 Hence, to get a correct hypothesis for the general form of the solution
to a boundary-value problem for a partial difference equation we have
to solve first several first-order linear difference equations.
First three or four equations seems will be always enough for doing this.
\end{remark}


In the following two corollaries we have that
$f_n=n$ for $n\in\mathbb{N}$,
which is one of the cases that is naturally appeared in several subareas
of mathematics.


\begin{corollary} \label{coro1}
The  boundary-value problem
\begin{equation}
\begin{gathered}
d_{m,n}=d_{m-1,n}+nd_{m-1,n-1},\quad 1\le n<m,\\
d_{m,0}=1,\quad d_{m,m}=m!,\quad m\in\mathbb{N}
\end{gathered} \label{bvc}
\end{equation}
has solution
\begin{equation}
d_{m,n}=\prod_{j=1}^n(m-n+j),\label{f0w}
\end{equation}
for every $(m,n)\in \mathcal{C}$.
\end{corollary}

\begin{proof}
 Using formula \eqref{a20} we have
\begin{equation}
\begin{aligned}
d_{m,n}
&=\sum_{j=1}^nC^{m-j-1}_{m-n-1}j!
 \prod_{s=j+1}^ns+\sum_{j=1}^{m-n}C^{m-1-j}_{m-n-j}\prod_{s=1}^ns\\
&=n!\Big(\sum_{j=1}^nC^{m-j-1}_{m-n-1}
 +\sum_{j=1}^{m-n}C^{m-1-j}_{m-n-j}\Big),
\end{aligned} \label{f1w}
\end{equation}
for every $(m,n)\in \mathcal{C}$.
We have
\begin{equation}
 \sum_{j=1}^nC^{m-j-1}_{m-n-1}
=\sum_{j=1}^n\big(C^{m-j}_{m-n}-C^{m-j-1}_{m-n})
=C^{m-1}_{m-n}-C^{m-n-1}_{m-n}
=C^{m-1}_{m-n},\label{f2w}
\end{equation}
and
\begin{equation}
 \sum_{j=1}^{m-n}C^{m-1-j}_{m-n-j}
=\sum_{j=1}^{m-n}C^{m-1-j}_{n-1}
=\sum_{j=1}^{m-n}\big(C^{m-j}_n-C^{m-1-j}_n\big)
=C^{m-1}_{m-n-1}.\label{f3w}
\end{equation}
From \eqref{f1w}-\eqref{f3w} we obtain
\[
d_{m,n}=n!\big(C^{m-1}_{m-n}+C^{m-1}_{m-n-1}\big)
=n!C^m_{m-n}=n!C^m_n
=m(m-1)\cdots(m-n+1),
\]
which is nothing but formula \eqref{f0w}.
\end{proof}


\begin{corollary} \label{coro2}
The boundary-value problem
\begin{equation}
\begin{gathered}
d_{m,n}=d_{m-1,n}+nd_{m-1,n-1},\quad 1\le n<m,\\
d_{m,0}=0,\quad d_{m,m}=m!,\quad m\in\mathbb{N}.
\end{gathered} \label{bvcw}
\end{equation}
has solution
\begin{equation}
d_{m,n}=n\prod_{j=1}^{n-1}(m-n+j).\label{f0ww}
\end{equation}
\end{corollary}

\begin{proof}
 Using formula \eqref{a20}, then \eqref{f2w}, and finally the symmetry of
the binomial coefficients, we have
\begin{equation}
\begin{aligned}
d_{m,n}
&=\sum_{j=1}^nC^{m-j-1}_{m-n-1}j!\prod_{s=j+1}^ns
 +\sum_{j=1}^{m-n}C^{m-1-j}_{m-n-j}\cdot 0\cdot \prod_{s=1}^ns\\
&=n!\sum_{j=1}^nC^{m-j-1}_{m-n-1}\\
&=n!C^{m-1}_{m-n}=n!C^{m-1}_{n-1}.
\end{aligned} \label{c1}
\end{equation}
From \eqref{c1}, formula \eqref{f0ww} easily follows.
\end{proof}

\begin{remark} \label{rmk2} \rm
 By choosing the sequences $(a_m)_{m\in\mathbb{N}}$, $(b_m)_{m\in\mathbb{N}}$
and $(f_n)_{n\in\mathbb{N}}$ at will, it can be obtained various other
interesting formulas for solutions to the boundary-value problem
 \eqref{a3}-\eqref{a3a}. As we have already mentioned, sequence $f_n$
can be chosen to have (arbitrary many) zeros. We leave it to the
imagination of the reader.
\end{remark}


\begin{thebibliography}{00}

\bibitem{ra} R. P. Agarwal;
\emph{Difference Equations and Inequalities: Theory, Methods, and Applications}
2nd Edition, Marcel Dekker Inc., New York, Basel, 2000.

\bibitem{ap} R. P. Agarwal, J. Popenda;
 Periodic solutions of first order linear difference equations,
\emph{Math. Comput. Modelling}, \textbf{22} (1) (1995), 11-19.

\bibitem{am} A. Ashyralyev, M. Modanli;
 An operator method for telegraph partial differential and difference equations,
\emph{Bound. Value Probl.}, Vol. 2015, Article No. 41, (2015), 17 pages.

\bibitem{bms} L. Berezansky, M. Migda, E. Schmeidel;
 Some stability conditions for scalar Volterra difference equations,
\emph{Opuscula Math.}, \textbf{36} (4) (2016), 459-470.

\bibitem{amc218-sde} L. Berg, S. Stevi\'c;
 On some systems of difference equations,
\emph{Appl. Math. Comput.}, \textbf{218} (2011), 1713-1718.

\bibitem{lb1} L. Brand;
 A sequence defined by a difference equation, \emph{Amer. Math. Monthly},
 \textbf{62} (7) (1955), 489-492.

\bibitem{lb} L. Brand;
 \emph{Differential and Difference Equations}, John Wiley $\&$ Sons,
Inc. New York, 1966.

\bibitem{ch} S. S. Cheng;
\emph{Partial Difference Equations},
Taylor $\&$ Francis, London and New York, 2003.

\bibitem{chu} S. S. Cheng, Y. F. Lu;
 General solutions of a three-level partial difference equation,
 \emph{Comput. Math. Appl.}, \textbf{38} (7-8) (1999), 65-79.

\bibitem{j} C. Jordan;
 \emph{Calculus of Finite Differences}, Chelsea Publishing Company, New York, 1956.

\bibitem{k} V. A. Krechmar;
 \emph{A Problem Book in Algebra}, Mir Publishers, Moscow, 1974.

\bibitem{ll} H. Levy, F. Lessman;
\emph{Finite Difference Equations}, Dover Publications, Inc., New York, 1992.

\bibitem{mm} M. Malin;
 Multiple solutions for a class of oscillatory discrete problems,
\emph{Adv. Nonlinear Anal.}, \textbf{4} (3) (2015), 221-233.

\bibitem{mk1} D. S. Mitrinovi\'c;
\emph{Mathematical Induction, Binomial Formula, Combinatorics},
 Gradjevinska Knjiga, Beograd, 1980 (in Serbian).

\bibitem{mk} D. S. Mitrinovi\'c, J. D. Ke\v{c}ki\'c;
\emph{Methods for Calculating Finite Sums}, Nau\v{c}na Knjiga, Beograd,
1984 (in Serbian).

\bibitem{ps1} G. Papaschinopoulos, C. J. Schinas;
 On a system of two nonlinear difference equations,
\emph{J. Math. Anal. Appl.}, \textbf{219} (2) (1998), 415-426.

\bibitem{ps2} G. Papaschinopoulos, C. J. Schinas;
On the behavior of the solutions of a system
of two nonlinear difference equations, \emph{Comm. Appl. Nonlinear
Anal.}, \textbf{5} (2) (1998), 47-59.

\bibitem{ps3} G. Papaschinopoulos, C. J. Schinas;
Invariants for systems of two nonlinear difference equations,
\emph{Differential Equations Dynam. Systems} \textbf{7} (2), (1999), 181-196.

\bibitem{pst2} G. Papaschinopoulos, G. Stefanidou;
Asymptotic behavior of the solutions of a class of rational difference equations,
\emph{Inter. J. Difference Equations}, \textbf{5} (2) (2010), 233-249.

\bibitem{rv} V. D. R\u{a}dulescu;
 Nonlinear elliptic equations with variable exponent: old and new,
\emph{Nonlinear Anal.}, \textbf{121} (2015), 336-369.

\bibitem{rr1} V. R\u{a}dulescu, D. Repov\v s;
\emph{Partial Differential Equations with Variable Exponents:
Variational Methods and Qualitative Analysis,}
CRC Press, Taylor and Francis Group, Boca Raton FL, 2015.

\bibitem{ri} J. Riordan;
 \emph{Combinatorial Identities}, John Wiley \& Sons Inc., New
York-London-Sydney, 1968.

\bibitem{amc219-sstho} S. Stevi\'c;
 On the system of difference equations
$x_n=c_ny_{n-3}/(a_n+b_ny_{n-1}x_{n-2}y_{n-3})$,
$y_n=\gamma_n x_{n-3}/(\alpha_n+\beta_n x_{n-1}y_{n-2}x_{n-3})$, \emph{Appl.
Math. Comput.}, \textbf{219} (2013), 4755-4764.

\bibitem{amc219-ssktho} S. Stevi\'c;
 On the system $x_{n+1}=y_nx_{n-k}/(y_{n-k+1}(a_n+b_ny_nx_{n-k}))$,
$y_{n+1}=x_ny_{n-k}/(x_{n-k+1}(c_n+d_nx_ny_{n-k}))$, \emph{Appl.
Math. Comput.}, \textbf{219} (2013), 4526-4534.

\bibitem{ejqtde-pdce} S. Stevi\'c;
 Note on the binomial partial difference equation, \emph{Electron. J. Qual. Theory
Differ. Equ.}, Vol. 2015, Article No. 96, (2015), 11 pages.

\bibitem{ade-2016pdce} S. Stevi\'c;
 Solvability of boundary value problems for a class of partial difference
equations on the combinatorial domain, \emph{Adv. Difference Equ.}, Vol. 2016,
Article No. 262, (2016), 10 pages.

\bibitem{ana1} S. Stevi\'c;
Solvable subclasses of a class of nonlinear second-order difference equations,
\emph{Adv. Nonlinear Anal.}, \textbf{5} (2) (2016), 147-165.

\bibitem{jdea205} S. Stevi\'c, J. Diblik, B. Iri\v{c}anin, Z. \v Smarda;
 On a solvable system of rational difference equations,
\emph{J. Difference Equ. Appl.}, \textbf{20} (5-6) (2014), 811-825.

\bibitem{ejde1} S. Stevi\'c, J. Diblik, B. Iri\v{c}anin, Z. \v Smarda;
 Solvability of nonlinear difference equations of
fourth order, \emph{Electron. J. Differential Equations}, Vol. 2014,
Article No. 264, (2014), 14 pages.

\bibitem{ade1} S. Stevi\'c, B. Iri\v{c}anin, Z. \v Smarda;
 On a close to symmetric system of difference
equations of second order, \emph{Adv. Difference Equ.}, Vol. 2015,
Article No. 264, (2015), 17 pages.

\bibitem{ya} S. V. Yablonskiy;
\emph{Introduction to Discrete Mathematics}, Mir Publishers, Moscow, 1989.


\end{thebibliography}



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