Theory and Modern Applications

# Discrete matrix delayed exponential for two delays and its property

## Abstract

In recent papers, a discrete matrix delayed exponential for a single delay was defined and its main property connected with the solution of linear discrete systems with a single delay was proved. In the present paper, a generalization of the concept of discrete matrix delayed exponential is designed for two delays and its main property is proved as well.

## Introduction

Throughout the paper, we use the following notation. For integers s, t, $s\le t$, we define a set ${\mathbb{Z}}_{s}^{t}:=\left\{s,s+1,\dots ,t-1,t\right\}$. Similarly, we define sets ${\mathbb{Z}}_{-\mathrm{\infty }}^{t}:=\left\{\dots ,t-1,t\right\}$ and ${\mathbb{Z}}_{s}^{\mathrm{\infty }}:=\left\{s,s+1,\dots \right\}$. The function $⌊\cdot ⌋$ used below is the floor integer function. We employ the following property of the floor integer function:

$x-1<⌊x⌋\le x,$
(1)

where $x\in \mathbb{R}$.

Define binomial coefficients as customary, i.e., for $n\in \mathbb{Z}$ and $k\in \mathbb{Z}$,

(2)

We recall that for a well-defined discrete function $f\left(k\right)$, the forward difference operator Δ is defined as $\mathrm{\Delta }f\left(k\right)=f\left(k+1\right)-f\left(k\right)$. In the paper, we also adopt the customary notation ${\sum }_{i={i}_{1}}^{{i}_{2}}{g}_{i}=0$ if ${i}_{2}<{i}_{1}$. In the case of double sums, we set

$\sum _{i={i}_{1},j={j}_{1}}^{{i}_{2},{j}_{2}}{g}_{ij}=0$
(3)

if at least one of the inequalities ${i}_{2}<{i}_{1}$, ${j}_{2}<{j}_{1}$ holds.

In [1, 2], a discrete matrix delayed exponential for a single delay $m\in \mathbb{N}$ was defined as follows.

Definition 1 For an $r×r$ constant matrix B, $k\in \mathbb{Z}$, and fixed $m\in \mathbb{N}$, we define the discrete matrix delayed exponential ${\mathrm{e}}_{m}^{Bk}$ as follows:

where Θ is an $r×r$ null matrix and I is an $r×r$ unit matrix.

Next, the main property (Theorem 1 below) of discrete matrix delayed exponential for a single delay $m\in \mathbb{N}$ is proved in .

Theorem 1 Let B be a constant $r×r$ matrix. Then, for $k\in {\mathbb{Z}}_{-m}^{\mathrm{\infty }}$,

$\mathrm{\Delta }{\mathrm{e}}_{m}^{Bk}=B{\mathrm{e}}_{m}^{B\left(k-m\right)}.$
(4)

The paper is concerned with a generalization of the notion of discrete matrix delayed exponential for two delays and a proof of one of its properties, similar to the main property (4) of discrete matrix delayed exponential for a single delay.

## Discrete matrix delayed exponential for two delays and its main property

We define a discrete $r×r$ matrix function ${\mathrm{e}}_{mn}^{BCk}$ called the discrete matrix delayed exponential for two delays $m,n\in \mathbb{N}$, $m\ne n$ and for two $r×r$ commuting constant matrices B, C as follows.

Definition 2 Let B, C be constant $r×r$ matrices with the property $BC=CB$ and let $m,n\in \mathbb{N}$, $m\ne n$ be fixed integers. We define a discrete $r×r$ matrix function ${\mathrm{e}}_{mn}^{BCk}$ called the discrete matrix delayed exponential for two delays m, n and for two $r×r$ constant matrices B, C:

where

${p}_{\left(k\right)}:=⌊\frac{k+m}{m+1}⌋,\phantom{\rule{2em}{0ex}}{q}_{\left(k\right)}:=⌊\frac{k+n}{n+1}⌋.$
(5)

The main property of ${\mathrm{e}}_{mn}^{BCk}$ is given by the following theorem.

Theorem 2 Let B, C be constant $r×r$ matrices with the property $BC=CB$ and let $m,n\in \mathbb{N}$, $m\ne n$ be fixed integers. Then

$\mathrm{\Delta }{\mathrm{e}}_{mn}^{BCk}=B{\mathrm{e}}_{mn}^{BC\left(k-m\right)}+C{\mathrm{e}}_{mn}^{BC\left(k-n\right)}$
(6)

holds for $k\ge 0$.

Proof Let $k\ge 1$. From (1) and (5), we can see easily that, for an integer $k\ge 0$ satisfying

$\left({p}_{\left(k\right)}-1\right)\left(m+1\right)+1\le k\le {p}_{\left(k\right)}\left(m+1\right)\wedge \left({q}_{\left(k\right)}-1\right)\left(n+1\right)+1\le k\le {q}_{\left(k\right)}\left(n+1\right),$

the relation

$\mathrm{\Delta }{\mathrm{e}}_{mn}^{BCk}=\mathrm{\Delta }\left[I+\left(B+C\right)\sum _{i=0,j=0}^{{p}_{\left(k\right)}-1,{q}_{\left(k\right)}-1}{B}^{i}{C}^{j}\left(\genfrac{}{}{0}{}{i+j}{i}\right)\left(\genfrac{}{}{0}{}{k-mi-nj}{i+j+1}\right)\right]$

holds in accordance with Definition 2 of ${\mathrm{e}}_{mn}^{BCk}$. Since $\mathrm{\Delta }I=\mathrm{\Theta }$, we have

$\mathrm{\Delta }{\mathrm{e}}_{mn}^{BCk}=\mathrm{\Delta }\left[\left(B+C\right)\sum _{i=0,j=0}^{{p}_{\left(k\right)}-1,{q}_{\left(k\right)}-1}{B}^{i}{C}^{j}\left(\genfrac{}{}{0}{}{i+j}{i}\right)\left(\genfrac{}{}{0}{}{k-mi-nj}{i+j+1}\right)\right].$
(7)

Considering the increment by its definition, i.e.,

$\mathrm{\Delta }{\mathrm{e}}_{mn}^{BCk}={\mathrm{e}}_{mn}^{BC\left(k+1\right)}-{\mathrm{e}}_{mn}^{BCk},$
(8)

we conclude that it is reasonable to divide the proof into four parts with respect to the value of integer k. In case one, k is such that

$\left({p}_{\left(k\right)}-1\right)\left(m+1\right)+1\le k<{p}_{\left(k\right)}\left(m+1\right)\wedge \left({q}_{\left(k\right)}-1\right)\left(n+1\right)+1\le k<{q}_{\left(k\right)}\left(n+1\right),$

in case two

$k={p}_{\left(k\right)}\left(m+1\right)\wedge \left({q}_{\left(k\right)}-1\right)\left(n+1\right)+1\le k<{q}_{\left(k\right)}\left(n+1\right),$

in case three

$\left({p}_{\left(k\right)}-1\right)\left(m+1\right)+1\le k<{p}_{\left(k\right)}\left(m+1\right)\wedge k={q}_{\left(k\right)}\left(n+1\right)$

and in case four

$k={p}_{\left(k\right)}\left(m+1\right)\wedge k={q}_{\left(k\right)}\left(n+1\right).$

We see that the above cases cover all the possible relations between k, ${p}_{\left(k\right)}$ and ${q}_{\left(k\right)}$.

In the proof, we use the identities

$\left(\genfrac{}{}{0}{}{n+1}{k}\right)=\left(\genfrac{}{}{0}{}{n}{k}\right)+\left(\genfrac{}{}{0}{}{n}{k-1}\right),$
(9)

where $n,k\in \mathbb{N}$ and

$\left(\genfrac{}{}{0}{}{i}{i}\right)=\left(\genfrac{}{}{0}{}{i-1}{i-1}\right),\phantom{\rule{2em}{0ex}}\left(\genfrac{}{}{0}{}{j}{0}\right)=\left(\genfrac{}{}{0}{}{j-1}{0}\right),\phantom{\rule{2em}{0ex}}\left(\genfrac{}{}{0}{}{i+j}{i}\right)=\left(\genfrac{}{}{0}{}{i+j-1}{i-1}\right)+\left(\genfrac{}{}{0}{}{i+j-1}{i}\right),$
(10)

where $i,j\in \mathbb{N}$, which are derived from (2) and (9).

### I. $\left({p}_{\left(k\right)}-1\right)\left(m+1\right)+1\le k<{p}_{\left(k\right)}\left(m+1\right)\wedge \left({q}_{\left(k\right)}-1\right)\left(n+1\right)+1\le k<{q}_{\left(k\right)}\left(n+1\right)$

From (1) and (5), we get

$\begin{array}{r}{p}_{\left(k-m\right)}=⌊\frac{k-m+m}{m+1}⌋\le \frac{k}{m+1}<{p}_{\left(k\right)},\\ {p}_{\left(k-m\right)}=⌊\frac{k-m+m}{m+1}⌋>\frac{k}{m+1}-1=\frac{k-m-1}{m+1}>{p}_{\left(k\right)}-2.\end{array}$

Therefore, ${p}_{\left(k-m\right)}={p}_{\left(k\right)}-1$ and, by Definition 2,

${\mathrm{e}}_{mn}^{BC\left(k-m\right)}=I+\left(B+C\right)\sum _{i=0,j=0}^{{p}_{\left(k\right)}-2,{q}_{\left(k-m\right)}-1}{B}^{i}{C}^{j}\left(\genfrac{}{}{0}{}{i+j}{i}\right)\left(\genfrac{}{}{0}{}{k-m-mi-nj}{i+j+1}\right).$
(11)

Similarly, omitting details, we get (using (1), and (5)) ${q}_{\left(k-n\right)}={q}_{\left(k\right)}-1$ and

${\mathrm{e}}_{mn}^{BC\left(k-n\right)}=I+\left(B+C\right)\sum _{i=0,j=0}^{{p}_{\left(k-n\right)}-1,{q}_{\left(k\right)}-2}{B}^{i}{C}^{j}\left(\genfrac{}{}{0}{}{i+j}{i}\right)\left(\genfrac{}{}{0}{}{k-n-mi-nj}{i+j+1}\right).$
(12)

Let ${q}_{\left(k-m\right)}\ge 1$. We show that

(13)

In accordance with (1),

${q}_{\left(k-m\right)}=⌊\frac{k-m+n}{n+1}⌋>\frac{k-m+n}{n+1}-1=\frac{k-m-1}{n+1}$

or

From the last inequality, we get

and (13) holds by (2). For that reason and since ${q}_{\left(k-m\right)}\le {q}_{\left(k\right)}$, we can replace ${q}_{\left(k-m\right)}$ by ${q}_{\left(k\right)}$ in (11). Thus, we have

${\mathrm{e}}_{mn}^{BC\left(k-m\right)}=I+\left(B+C\right)\sum _{i=0,j=0}^{{p}_{\left(k\right)}-2,{q}_{\left(k\right)}-1}{B}^{i}{C}^{j}\left(\genfrac{}{}{0}{}{i+j}{i}\right)\left(\genfrac{}{}{0}{}{k-m\left(i+1\right)-nj}{i+j+1}\right).$
(14)

It is easy to see that, due to (3), formula (14) can be used instead of (11) if ${q}_{\left(k-m\right)}<1$ also.

Let ${p}_{\left(k-n\right)}\ge 1$. Similarly, we can show that

and, since ${p}_{\left(k-n\right)}\le {p}_{\left(k\right)}$, we can replace ${p}_{\left(k-n\right)}$ by ${p}_{\left(k\right)}$ in (12). Thus, we have

${\mathrm{e}}_{mn}^{BC\left(k-n\right)}=I+\left(B+C\right)\sum _{i=0,j=0}^{{p}_{\left(k\right)}-1,{q}_{\left(k\right)}-2}{B}^{i}{C}^{j}\left(\genfrac{}{}{0}{}{i+j}{i}\right)\left(\genfrac{}{}{0}{}{k-mi-n\left(j+1\right)}{i+j+1}\right).$
(15)

It is easy to see that, due to (3), formula (15) can be used instead of (12) if ${p}_{\left(k-n\right)}<1$, too.

Due to (1), we also conclude that

${p}_{\left(k+1\right)}={p}_{\left(k\right)},\phantom{\rule{2em}{0ex}}{q}_{\left(k+1\right)}={q}_{\left(k\right)}$
(16)

because

${p}_{\left(k+1\right)}=⌊\frac{k+1+m}{m+1}⌋\le \frac{k}{m+1}+1<{p}_{\left(k\right)}+1$

and

${p}_{\left(k+1\right)}=⌊\frac{k+1+m}{m+1}⌋>\frac{k+1+m}{m+1}-1=\frac{k}{m+1}\ge {p}_{\left(k\right)}-1+\frac{1}{m+1}.$

The second formula can be proved similarly.

Now we are able to prove that

$\begin{array}{rl}\mathrm{\Delta }{\mathrm{e}}_{mn}^{BCk}=& B{\mathrm{e}}_{mn}^{BC\left(k-m\right)}+C{\mathrm{e}}_{mn}^{BC\left(k-n\right)}\\ =& B\left[I+\left(B+C\right)\sum _{i=0,j=0}^{{p}_{\left(k\right)}-2,{q}_{\left(k\right)}-1}{B}^{i}{C}^{j}\left(\genfrac{}{}{0}{}{i+j}{i}\right)\left(\genfrac{}{}{0}{}{k-m\left(i+1\right)-nj}{i+j+1}\right)\right]\\ +C\left[I+\left(B+C\right)\sum _{i=0,j=0}^{{p}_{\left(k\right)}-1,{q}_{\left(k\right)}-2}{B}^{i}{C}^{j}\left(\genfrac{}{}{0}{}{i+j}{i}\right)\left(\genfrac{}{}{0}{}{k-mi-n\left(j+1\right)}{i+j+1}\right)\right].\end{array}$
(17)

With the aid of (7), (8), (9) and (16), we get

$\begin{array}{rl}\mathrm{\Delta }{\mathrm{e}}_{mn}^{BCk}=& {\mathrm{e}}_{mn}^{BC\left(k+1\right)}-{\mathrm{e}}_{mn}^{BCk}\\ =& I+\left(B+C\right)\sum _{i=0,j=0}^{{p}_{\left(k+1\right)}-1,{q}_{\left(k+1\right)}-1}{B}^{i}{C}^{j}\left(\genfrac{}{}{0}{}{i+j}{i}\right)\left(\genfrac{}{}{0}{}{k+1-mi-nj}{i+j+1}\right)\\ -I-\left(B+C\right)\sum _{i=0,j=0}^{{p}_{\left(k\right)}-1,{q}_{\left(k\right)}-1}{B}^{i}{C}^{j}\left(\genfrac{}{}{0}{}{i+j}{i}\right)\left(\genfrac{}{}{0}{}{k-mi-nj}{i+j+1}\right)\\ =& I+\left(B+C\right)\sum _{i=0,j=0}^{{p}_{\left(k\right)}-1,{q}_{\left(k\right)}-1}{B}^{i}{C}^{j}\left(\genfrac{}{}{0}{}{i+j}{i}\right)\left(\genfrac{}{}{0}{}{k+1-mi-nj}{i+j+1}\right)\\ -I-\left(B+C\right)\sum _{i=0,j=0}^{{p}_{\left(k\right)}-1,{q}_{\left(k\right)}-1}{B}^{i}{C}^{j}\left(\genfrac{}{}{0}{}{i+j}{i}\right)\left(\genfrac{}{}{0}{}{k-mi-nj}{i+j+1}\right)\\ =& \left(B+C\right)\sum _{i=0,j=0}^{{p}_{\left(k\right)}-1,{q}_{\left(k\right)}-1}{B}^{i}{C}^{j}\left(\genfrac{}{}{0}{}{i+j}{i}\right)\left[\left(\genfrac{}{}{0}{}{k+1-mi-nj}{i+j+1}\right)-\left(\genfrac{}{}{0}{}{k-mi-nj}{i+j+1}\right)\right]\\ =& \left(B+C\right)\sum _{i=0,j=0}^{{p}_{\left(k\right)}-1,{q}_{\left(k\right)}-1}{B}^{i}{C}^{j}\left(\genfrac{}{}{0}{}{i+j}{i}\right)\left(\genfrac{}{}{0}{}{k-mi-nj}{i+j}\right)\\ =& \left(B+C\right)\left[I+\sum _{i=1}^{{p}_{\left(k\right)}-1}{B}^{i}{C}^{0}\left(\genfrac{}{}{0}{}{i}{i}\right)\left(\genfrac{}{}{0}{}{k-mi}{i}\right)+\sum _{j=1}^{{q}_{\left(k\right)}-1}{B}^{0}{C}^{j}\left(\genfrac{}{}{0}{}{j}{0}\right)\left(\genfrac{}{}{0}{}{k-nj}{j}\right)\\ +\sum _{i=1,j=1}^{{p}_{\left(k\right)}-1,{q}_{\left(k\right)}-1}{B}^{i}{C}^{j}\left(\genfrac{}{}{0}{}{i+j}{i}\right)\left(\genfrac{}{}{0}{}{k-mi-nj}{i+j}\right)\right].\end{array}$

By (10), we have

$\begin{array}{rl}\mathrm{\Delta }{\mathrm{e}}_{mn}^{BCk}=& \left(B+C\right)\left[I+\sum _{i=1}^{{p}_{\left(k\right)}-1}{B}^{i}{C}^{0}\left(\genfrac{}{}{0}{}{i-1}{i-1}\right)\left(\genfrac{}{}{0}{}{k-mi}{i}\right)\\ +\sum _{j=1}^{{q}_{\left(k\right)}-1}{B}^{0}{C}^{j}\left(\genfrac{}{}{0}{}{j-1}{0}\right)\left(\genfrac{}{}{0}{}{k-nj}{j}\right)\\ +\sum _{i=1,j=1}^{{p}_{\left(k\right)}-1,{q}_{\left(k\right)}-1}{B}^{i}{C}^{j}\left(\genfrac{}{}{0}{}{i+j-1}{i-1}\right)\left(\genfrac{}{}{0}{}{k-mi-nj}{i+j}\right)\\ +\sum _{i=1,j=1}^{{p}_{\left(k\right)}-1,{q}_{\left(k\right)}-1}{B}^{i}{C}^{j}\left(\genfrac{}{}{0}{}{i+j-1}{i}\right)\left(\genfrac{}{}{0}{}{k-mi-nj}{i+j}\right)\right]\\ =& \left(B+C\right)\left[I+\sum _{i=1,j=0}^{{p}_{\left(k\right)}-1,{q}_{\left(k\right)}-1}{B}^{i}{C}^{j}\left(\genfrac{}{}{0}{}{i+j-1}{i-1}\right)\left(\genfrac{}{}{0}{}{k-mi-nj}{i+j}\right)\\ +\sum _{i=0,j=1}^{{p}_{\left(k\right)}-1,{q}_{\left(k\right)}-1}{B}^{i}{C}^{j}\left(\genfrac{}{}{0}{}{i+j-1}{i}\right)\left(\genfrac{}{}{0}{}{k-mi-nj}{i+j}\right)\right].\end{array}$

Now in the first sum we replace the summation index i by $i+1$ and in the second sum we replace the summation index j by $j+1$. Then

$\begin{array}{rl}\mathrm{\Delta }{\mathrm{e}}_{mn}^{BCk}=& \left(B+C\right)\left[I+\sum _{i=0,j=0}^{{p}_{\left(k\right)}-2,{q}_{\left(k\right)}-1}{B}^{i+1}{C}^{j}\left(\genfrac{}{}{0}{}{i+j}{i}\right)\left(\genfrac{}{}{0}{}{k-m\left(i+1\right)-nj}{i+j+1}\right)\\ +\sum _{i=0,j=0}^{{p}_{\left(k\right)}-1,{q}_{\left(k\right)}-2}{B}^{i}{C}^{j+1}\left(\genfrac{}{}{0}{}{i+j}{i}\right)\left(\genfrac{}{}{0}{}{k-mi-n\left(j+1\right)}{i+j+1}\right)\right]\\ =& B+B\left(B+C\right)\sum _{i=0,j=0}^{{p}_{\left(k\right)}-2,{q}_{\left(k\right)}-1}{B}^{i}{C}^{j}\left(\genfrac{}{}{0}{}{i+j}{i}\right)\left(\genfrac{}{}{0}{}{k-m\left(i+1\right)-nj}{i+j+1}\right)\\ +C+C\left(B+C\right)\sum _{i=0,j=0}^{{p}_{\left(k\right)}-1,{q}_{\left(k\right)}-2}{B}^{i}{C}^{j}\left(\genfrac{}{}{0}{}{i+j}{i}\right)\left(\genfrac{}{}{0}{}{k-mi-n\left(j+1\right)}{i+j+1}\right)\\ =& B\left[I+\left(B+C\right)\sum _{i=0,j=0}^{{p}_{\left(k\right)}-2,{q}_{\left(k\right)}-1}{B}^{i}{C}^{j}\left(\genfrac{}{}{0}{}{i+j}{i}\right)\left(\genfrac{}{}{0}{}{k-m\left(i+1\right)-nj}{i+j+1}\right)\right]\\ +C\left[I+C\left(B+C\right)\sum _{i=0,j=0}^{{p}_{\left(k\right)}-1,{q}_{\left(k\right)}-2}{B}^{i}{C}^{j}\left(\genfrac{}{}{0}{}{i+j}{i}\right)\left(\genfrac{}{}{0}{}{k-mi-n\left(j+1\right)}{i+j+1}\right)\right]\\ =& B{\mathrm{e}}_{mn}^{BC\left(k-m\right)}+C{\mathrm{e}}_{mn}^{BC\left(k-n\right)}.\end{array}$

Due to (14) and (15), we conclude that formula (17) is valid.

### II. $k={p}_{\left(k\right)}\left(m+1\right)\wedge \left({q}_{\left(k\right)}-1\right)\left(n+1\right)+1\le k<{q}_{\left(k\right)}\left(n+1\right)$

In this case,

$\begin{array}{r}{p}_{\left(k-m\right)}=⌊\frac{k-m+m}{m+1}⌋=⌊\frac{k}{m+1}⌋={p}_{\left(k\right)},\\ {p}_{\left(k+1\right)}=⌊\frac{k+1+m}{m+1}⌋\le \frac{k+1+m}{m+1}=\frac{k}{m+1}+1={p}_{\left(k\right)}+1,\\ {p}_{\left(k+1\right)}=⌊\frac{k+1+m}{m+1}⌋>\frac{k+1+m}{m+1}-1=\frac{k}{m+1}={p}_{\left(k\right)}\end{array}$

and ${p}_{\left(k+1\right)}={p}_{\left(k\right)}+1$. In addition to this (see relevant computations performed in case I), we have ${q}_{\left(k-n\right)}={q}_{\left(k\right)}-1$ and ${q}_{\left(k+1\right)}={q}_{\left(k\right)}$.

Then

$\begin{array}{r}{\mathrm{e}}_{mn}^{BCk}=I+\left(B+C\right)\sum _{i=0,j=0}^{{p}_{\left(k\right)}-1,{q}_{\left(k\right)}-1}{B}^{i}{C}^{j}\left(\genfrac{}{}{0}{}{i+j}{i}\right)\left(\genfrac{}{}{0}{}{k-mi-nj}{i+j+1}\right),\\ {\mathrm{e}}_{mn}^{BC\left(k+1\right)}=I+\left(B+C\right)\sum _{i=0,j=0}^{{p}_{\left(k\right)},{q}_{\left(k\right)}-1}{B}^{i}{C}^{j}\left(\genfrac{}{}{0}{}{i+j}{i}\right)\left(\genfrac{}{}{0}{}{k+1-mi-nj}{i+j+1}\right)\end{array}$

and

${\mathrm{e}}_{mn}^{BC\left(k-m\right)}=I+\left(B+C\right)\sum _{i=0,j=0}^{{p}_{\left(k\right)}-1,{q}_{\left(k-m\right)}-1}{B}^{i}{C}^{j}\left(\genfrac{}{}{0}{}{i+j}{i}\right)\left(\genfrac{}{}{0}{}{k-m-mi-nj}{i+j+1}\right),$
(18)
${\mathrm{e}}_{mn}^{BC\left(k-n\right)}=I+\left(B+C\right)\sum _{i=0,j=0}^{{p}_{\left(k-n\right)}-1,{q}_{\left(k\right)}-2}{B}^{i}{C}^{j}\left(\genfrac{}{}{0}{}{i+j}{i}\right)\left(\genfrac{}{}{0}{}{k-n-mi-nj}{i+j+1}\right).$
(19)

Like with the computations performed in the previous part of the proof, we get

and

So, we can substitute ${q}_{\left(k-m\right)}$ by ${q}_{\left(k\right)}$ in (18) and ${p}_{\left(k-n\right)}$ by ${p}_{\left(k\right)}$ in (19).

Accordingly, we have

${\mathrm{e}}_{mn}^{BC\left(k-m\right)}=I+\left(B+C\right)\sum _{i=0,j=0}^{{p}_{\left(k\right)}-1,{q}_{\left(k\right)}-1}{B}^{i}{C}^{j}\left(\genfrac{}{}{0}{}{i+j}{i}\right)\left(\genfrac{}{}{0}{}{k-m\left(i+1\right)-nj}{i+j+1}\right),$
(20)
${\mathrm{e}}_{mn}^{BC\left(k-n\right)}=I+\left(B+C\right)\sum _{i=0,j=0}^{{p}_{\left(k\right)}-1,{q}_{\left(k\right)}-2}{B}^{i}{C}^{j}\left(\genfrac{}{}{0}{}{i+j}{i}\right)\left(\genfrac{}{}{0}{}{k-mi-n\left(j+1\right)}{i+j+1}\right).$
(21)

It is easy to see that, due to (3), formula (20) can also be used instead of (18) if ${q}_{\left(k-m\right)}<1$ and formula (21) can also be used instead of (19) if ${p}_{\left(k-n\right)}<1$.

We have to prove

$\begin{array}{rl}\mathrm{\Delta }{\mathrm{e}}_{mn}^{BCk}=& B{\mathrm{e}}_{mn}^{BC\left(k-m\right)}+C{\mathrm{e}}_{mn}^{BC\left(k-n\right)}\\ =& B\left[I+\left(B+C\right)\sum _{i=0,j=0}^{{p}_{\left(k\right)}-1,{q}_{\left(k\right)}-1}{B}^{i}{C}^{j}\left(\genfrac{}{}{0}{}{i+j}{i}\right)\left(\genfrac{}{}{0}{}{k-m\left(i+1\right)-nj}{i+j+1}\right)\right]\\ +C\left[I+\left(B+C\right)\sum _{i=0,j=0}^{{p}_{\left(k\right)}-1,{q}_{\left(k\right)}-2}{B}^{i}{C}^{j}\left(\genfrac{}{}{0}{}{i+j}{i}\right)\left(\genfrac{}{}{0}{}{k-mi-n\left(j+1\right)}{i+j+1}\right)\right].\end{array}$
(22)

Therefore,

$\begin{array}{rl}\mathrm{\Delta }{\mathrm{e}}_{mn}^{BCk}=& {\mathrm{e}}_{mn}^{BC\left(k+1\right)}-{\mathrm{e}}_{mn}^{BCk}\\ =& I+\left(B+C\right)\sum _{i=0,j=0}^{{p}_{\left(k\right)},{q}_{\left(k\right)}-1}{B}^{i}{C}^{j}\left(\genfrac{}{}{0}{}{i+j}{i}\right)\left(\genfrac{}{}{0}{}{k+1-mi-nj}{i+j+1}\right)\\ -I-\left(B+C\right)\sum _{i=0,j=0}^{{p}_{\left(k\right)}-1,{q}_{\left(k\right)}-1}{B}^{i}{C}^{j}\left(\genfrac{}{}{0}{}{i+j}{i}\right)\left(\genfrac{}{}{0}{}{k-mi-nj}{i+j+1}\right)\\ =& \left(B+C\right)\left[\sum _{i=0,j=0}^{{p}_{\left(k\right)}-1,{q}_{\left(k\right)}-1}{B}^{i}{C}^{j}\left(\genfrac{}{}{0}{}{i+j}{i}\right)\left[\left(\genfrac{}{}{0}{}{k+1-mi-nj}{i+j+1}\right)-\left(\genfrac{}{}{0}{}{k-mi-nj}{i+j+1}\right)\right]\\ +\sum _{j=0}^{{q}_{\left(k\right)}-1}{B}^{{p}_{\left(k\right)}}{C}^{j}\left(\genfrac{}{}{0}{}{{p}_{\left(k\right)}+j}{{p}_{\left(k\right)}}\right)\left(\genfrac{}{}{0}{}{k+1-m{p}_{\left(k\right)}-nj}{{p}_{\left(k\right)}+j+1}\right)\right].\end{array}$

With the aid of the equation $k={p}_{\left(k\right)}\left(m+1\right)$, we get

and, by (9), we have

$\begin{array}{rl}\mathrm{\Delta }{\mathrm{e}}_{mn}^{BCk}=& \left(B+C\right)\left[\sum _{i=0,j=0}^{{p}_{\left(k\right)}-1,{q}_{\left(k\right)}-1}{B}^{i}{C}^{j}\left(\genfrac{}{}{0}{}{i+j}{i}\right)\left(\genfrac{}{}{0}{}{k-mi-nj}{i+j}\right)+{B}^{{p}_{\left(k\right)}}\right]\\ =& \left(B+C\right)\left[I+\sum _{i=1}^{{p}_{\left(k\right)}-1}{B}^{i}{C}^{0}\left(\genfrac{}{}{0}{}{i}{i}\right)\left(\genfrac{}{}{0}{}{k-mi}{i}\right)+\sum _{j=1}^{{q}_{\left(k\right)}-1}{B}^{0}{C}^{j}\left(\genfrac{}{}{0}{}{j}{0}\right)\left(\genfrac{}{}{0}{}{k-nj}{j}\right)\\ +\sum _{i=1,j=1}^{{p}_{\left(k\right)}-1,{q}_{\left(k\right)}-1}{B}^{i}{C}^{j}\left(\genfrac{}{}{0}{}{i+j}{i}\right)\left(\genfrac{}{}{0}{}{k-mi-nj}{i+j}\right)+{B}^{{p}_{\left(k\right)}}\right].\end{array}$

By (10), we have

$\begin{array}{rl}\mathrm{\Delta }{\mathrm{e}}_{mn}^{BCk}=& \left(B+C\right)\left[I+\sum _{i=1}^{{p}_{\left(k\right)}-1}{B}^{i}{C}^{0}\left(\genfrac{}{}{0}{}{i-1}{i-1}\right)\left(\genfrac{}{}{0}{}{k-mi}{i}\right)+\sum _{j=1}^{{q}_{\left(k\right)}-1}{B}^{0}{C}^{j}\left(\genfrac{}{}{0}{}{j-1}{0}\right)\left(\genfrac{}{}{0}{}{k-nj}{j}\right)\\ +\sum _{i=1,j=1}^{{p}_{\left(k\right)}-1,{q}_{\left(k\right)}-1}{B}^{i}{C}^{j}\left(\genfrac{}{}{0}{}{i+j-1}{i-1}\right)\left(\genfrac{}{}{0}{}{k-mi-nj}{i+j}\right)\\ +\sum _{i=1,j=1}^{{p}_{\left(k\right)}-1,{q}_{\left(k\right)}-1}{B}^{i}{C}^{j}\left(\genfrac{}{}{0}{}{i+j-1}{i}\right)\left(\genfrac{}{}{0}{}{k-mi-nj}{i+j}\right)+{B}^{{p}_{\left(k\right)}}\right]\\ =& \left(B+C\right)\left[I+\sum _{i=1,j=0}^{{p}_{\left(k\right)}-1,{q}_{\left(k\right)}-1}{B}^{i}{C}^{j}\left(\genfrac{}{}{0}{}{i+j-1}{i-1}\right)\left(\genfrac{}{}{0}{}{k-mi-nj}{i+j}\right)\\ +\sum _{i=0,j=1}^{{p}_{\left(k\right)}-1,{q}_{\left(k\right)}-1}{B}^{i}{C}^{j}\left(\genfrac{}{}{0}{}{i+j-1}{i}\right)\left(\genfrac{}{}{0}{}{k-mi-nj}{i+j}\right)+{B}^{{p}_{\left(k\right)}}\right].\end{array}$

Now we replace in the first sum the summation index i by $i+1$ and in the second sum we replace the summation index j by $j+1$. Then

$\begin{array}{rl}\mathrm{\Delta }{\mathrm{e}}_{mn}^{BCk}=& \left(B+C\right)\left[I+\sum _{i=0,j=0}^{{p}_{\left(k\right)}-2,{q}_{\left(k\right)}-1}{B}^{i+1}{C}^{j}\left(\genfrac{}{}{0}{}{i+j}{i}\right)\left(\genfrac{}{}{0}{}{k-m\left(i+1\right)-nj}{i+j+1}\right)\\ +\sum _{i=0,j=0}^{{p}_{\left(k\right)}-1,{q}_{\left(k\right)}-2}{B}^{i}{C}^{j+1}\left(\genfrac{}{}{0}{}{i+j}{i}\right)\left(\genfrac{}{}{0}{}{k-mi-n\left(j+1\right)}{i+j+1}\right)+{B}^{{p}_{\left(k\right)}}\right]\\ =& B+B\left(B+C\right)\sum _{i=0,j=0}^{{p}_{\left(k\right)}-2,{q}_{\left(k\right)}-1}{B}^{i}{C}^{j}\left(\genfrac{}{}{0}{}{i+j}{i}\right)\left(\genfrac{}{}{0}{}{k-m\left(i+1\right)-nj}{i+j+1}\right)+{B}^{{p}_{\left(k\right)}}\left(B+C\right)\\ +C+C\left(B+C\right)\sum _{i=0,j=0}^{{p}_{\left(k\right)}-1,{q}_{\left(k\right)}-2}{B}^{i}{C}^{j}\left(\genfrac{}{}{0}{}{i+j}{i}\right)\left(\genfrac{}{}{0}{}{k-mi-n\left(j+1\right)}{i+j+1}\right)\\ =& B\left[I+\left(B+C\right)\left(\sum _{i=0,j=0}^{{p}_{\left(k\right)}-2,{q}_{\left(k\right)}-1}{B}^{i}{C}^{j}\left(\genfrac{}{}{0}{}{i+j}{i}\right)\left(\genfrac{}{}{0}{}{k-m\left(i+1\right)-nj}{i+j+1}\right)+{B}^{{p}_{\left(k\right)}-1}\right)\right]\\ +C\left[I+\left(B+C\right)\sum _{i=0,j=0}^{{p}_{\left(k\right)}-1,{q}_{\left(k\right)}-2}{B}^{i}{C}^{j}\left(\genfrac{}{}{0}{}{i+j}{i}\right)\left(\genfrac{}{}{0}{}{k-mi-n\left(j+1\right)}{i+j+1}\right)\right].\end{array}$

For $k={p}_{\left(k\right)}\left(m+1\right)$, we have

${B}^{{p}_{\left(k\right)}-1}=\sum _{j=0}^{{q}_{\left(k\right)}-1}{B}^{{p}_{\left(k\right)}-1}{C}^{j}\left(\genfrac{}{}{0}{}{{p}_{\left(k\right)}-1+j}{{p}_{\left(k\right)}-1}\right)\left(\genfrac{}{}{0}{}{k-m\left({p}_{\left(k\right)}-1+1\right)-nj}{{p}_{\left(k\right)}-1+j+1}\right),$

where

Thus,

$\begin{array}{rl}\mathrm{\Delta }{\mathrm{e}}_{mn}^{BCk}=& B\left[I+\left(B+C\right)\sum _{i=0,j=0}^{{p}_{\left(k\right)}-1,{q}_{\left(k\right)}-1}{B}^{i}{C}^{j}\left(\genfrac{}{}{0}{}{i+j}{i}\right)\left(\genfrac{}{}{0}{}{k-m\left(i+1\right)-nj}{i+j+1}\right)\right]\\ +C\left[I+\left(B+C\right)\sum _{i=0,j=0}^{{p}_{\left(k\right)}-1,{q}_{\left(k\right)}-2}{B}^{i}{C}^{j}\left(\genfrac{}{}{0}{}{i+j}{i}\right)\left(\genfrac{}{}{0}{}{k-mi-n\left(j+1\right)}{i+j+1}\right)\right]\\ =& B{\mathrm{e}}_{mn}^{BC\left(k-m\right)}+C{\mathrm{e}}_{mn}^{BC\left(k-n\right)}\end{array}$

and formula (22) is proved.

### III. $\left({p}_{\left(k\right)}-1\right)\left(m+1\right)+1\le k<{p}_{\left(k\right)}\left(m+1\right)\wedge k={q}_{\left(k\right)}\left(n+1\right)$

In this case, we have (see relevant computations in cases I and II)

${p}_{\left(k-m\right)}={p}_{\left(k\right)}-1,\phantom{\rule{2em}{0ex}}{p}_{\left(k+1\right)}={p}_{\left(k\right)}$

and

${q}_{\left(k-n\right)}={q}_{\left(k\right)},\phantom{\rule{2em}{0ex}}{q}_{\left(k+1\right)}={q}_{\left(k\right)}+1.$

Then

$\begin{array}{r}{\mathrm{e}}_{mn}^{BCk}=I+\left(B+C\right)\sum _{i=0,j=0}^{{p}_{\left(k\right)}-1,{q}_{\left(k\right)}-1}{B}^{i}{C}^{j}\left(\genfrac{}{}{0}{}{i+j}{i}\right)\left(\genfrac{}{}{0}{}{k-mi-nj}{i+j+1}\right),\\ {\mathrm{e}}_{mn}^{BC\left(k+1\right)}=I+\left(B+C\right)\sum _{i=0,j=0}^{{p}_{\left(k\right)}-1,{q}_{\left(k\right)}}{B}^{i}{C}^{j}\left(\genfrac{}{}{0}{}{i+j}{i}\right)\left(\genfrac{}{}{0}{}{k+1-mi-nj}{i+j+1}\right)\end{array}$

and

${\mathrm{e}}_{mn}^{BC\left(k-m\right)}=I+\left(B+C\right)\sum _{i=0,j=0}^{{p}_{\left(k\right)}-2,{q}_{\left(k-m\right)}-1}{B}^{i}{C}^{j}\left(\genfrac{}{}{0}{}{i+j}{i}\right)\left(\genfrac{}{}{0}{}{k-m-mi-nj}{i+j+1}\right),$
(23)
${\mathrm{e}}_{mn}^{BC\left(k-n\right)}=I+\left(B+C\right)\sum _{i=0,j=0}^{{p}_{\left(k-n\right)}-1,{q}_{\left(k\right)}-1}{B}^{i}{C}^{j}\left(\genfrac{}{}{0}{}{i+j}{i}\right)\left(\genfrac{}{}{0}{}{k-n-mi-nj}{i+j+1}\right).$
(24)

Like with the computations performed in case I, we can get

and

So, we can substitute ${q}_{\left(k\right)}$ for ${q}_{\left(k-m\right)}$ in (23) and ${p}_{\left(k\right)}$ for ${p}_{\left(k-n\right)}$ in (24).

Thus, we have

${\mathrm{e}}_{mn}^{BC\left(k-m\right)}=I+\left(B+C\right)\sum _{i=0,j=0}^{{p}_{\left(k\right)}-2,{q}_{\left(k\right)}-1}{B}^{i}{C}^{j}\left(\genfrac{}{}{0}{}{i+j}{i}\right)\left(\genfrac{}{}{0}{}{k-m\left(i+1\right)-nj}{i+j+1}\right),$
(25)
${\mathrm{e}}_{mn}^{BC\left(k-n\right)}=I+\left(B+C\right)\sum _{i=0,j=0}^{{p}_{\left(k\right)}-1,{q}_{\left(k\right)}-1}{B}^{i}{C}^{j}\left(\genfrac{}{}{0}{}{i+j}{i}\right)\left(\genfrac{}{}{0}{}{k-mi-n\left(j+1\right)}{i+j+1}\right).$
(26)

It is easy to see that, due to (3), formula (25) can also be used instead of (23) if ${q}_{\left(k-m\right)}<1$ and formula (26) can also be used instead of (24) if ${p}_{\left(k-n\right)}<1$.

Now we have to prove

$\begin{array}{rl}\mathrm{\Delta }{\mathrm{e}}_{mn}^{BCk}=& B{\mathrm{e}}_{mn}^{BC\left(k-m\right)}+C{\mathrm{e}}_{mn}^{BC\left(k-n\right)}\\ =& B\left[I+\left(B+C\right)\sum _{i=0,j=0}^{{p}_{\left(k\right)}-2,{q}_{\left(k\right)}-1}{B}^{i}{C}^{j}\left(\genfrac{}{}{0}{}{i+j}{i}\right)\left(\genfrac{}{}{0}{}{k-m\left(i+1\right)-nj}{i+j+1}\right)\right]\\ +C\left[I+\left(B+C\right)\sum _{i=0,j=0}^{{p}_{\left(k\right)}-1,{q}_{\left(k\right)}-1}{B}^{i}{C}^{j}\left(\genfrac{}{}{0}{}{i+j}{i}\right)\left(\genfrac{}{}{0}{}{k-mi-n\left(j+1\right)}{i+j+1}\right)\right].\end{array}$
(27)

Considering the difference by its definition, we get

$\begin{array}{rl}\mathrm{\Delta }{\mathrm{e}}_{mn}^{BCk}=& {\mathrm{e}}_{mn}^{BC\left(k+1\right)}-{\mathrm{e}}_{mn}^{BCk}\\ =& I+\left(B+C\right)\sum _{i=0,j=0}^{{p}_{\left(k\right)}-1,{q}_{\left(k\right)}}{B}^{i}{C}^{j}\left(\genfrac{}{}{0}{}{i+j}{i}\right)\left(\genfrac{}{}{0}{}{k+1-mi-nj}{i+j+1}\right)\\ -I-\left(B+C\right)\sum _{i=0,j=0}^{{p}_{\left(k\right)}-1,{q}_{\left(k\right)}-1}{B}^{i}{C}^{j}\left(\genfrac{}{}{0}{}{i+j}{i}\right)\left(\genfrac{}{}{0}{}{k-mi-nj}{i+j+1}\right)\\ =& \left(B+C\right)\left[\sum _{i=0,j=0}^{{p}_{\left(k\right)}-1,{q}_{\left(k\right)}-1}{B}^{i}{C}^{j}\left(\genfrac{}{}{0}{}{i+j}{i}\right)\left[\left(\genfrac{}{}{0}{}{k+1-mi-nj}{i+j+1}\right)-\left(\genfrac{}{}{0}{}{k-mi-nj}{i+j+1}\right)\right]\\ +\sum _{i=0}^{{p}_{\left(k\right)}-1}{B}^{i}{C}^{{q}_{\left(k\right)}}\left(\genfrac{}{}{0}{}{i+{q}_{\left(k\right)}}{i}\right)\left(\genfrac{}{}{0}{}{k+1-mi-n{q}_{\left(k\right)}}{i+{q}_{\left(k\right)}+1}\right)\right].\end{array}$

With the aid of relation $k={q}_{\left(k\right)}\left(n+1\right)$, we get

and

$\begin{array}{rl}\mathrm{\Delta }{\mathrm{e}}_{mn}^{BCk}=& \left(B+C\right)\left[\sum _{i=0,j=0}^{{p}_{\left(k\right)}-1,{q}_{\left(k\right)}-1}{B}^{i}{C}^{j}\left(\genfrac{}{}{0}{}{i+j}{i}\right)\left(\genfrac{}{}{0}{}{k-mi-nj}{i+j}\right)+{C}^{{q}_{\left(k\right)}}\right]\\ =& \left(B+C\right)\left[I+\sum _{i=1}^{{p}_{\left(k\right)}-1}{B}^{i}{C}^{0}\left(\genfrac{}{}{0}{}{i}{i}\right)\left(\genfrac{}{}{0}{}{k-mi}{i}\right)\\ +\sum _{j=1}^{{q}_{\left(k\right)}-1}{B}^{0}{C}^{j}\left(\genfrac{}{}{0}{}{j}{0}\right)\left(\genfrac{}{}{0}{}{k-nj}{j}\right)\\ +\sum _{i=1,j=1}^{{p}_{\left(k\right)}-1,{q}_{\left(k\right)}-1}{B}^{i}{C}^{j}\left(\genfrac{}{}{0}{}{i+j}{i}\right)\left(\genfrac{}{}{0}{}{k-mi-nj}{i+j}\right)+{C}^{{q}_{\left(k\right)}}\right].\end{array}$

By (10), we have

$\begin{array}{rl}\mathrm{\Delta }{\mathrm{e}}_{mn}^{BCk}=& \left(B+C\right)\left[I+\sum _{i=1}^{{p}_{\left(k\right)}-1}{B}^{i}{C}^{0}\left(\genfrac{}{}{0}{}{i-1}{i-1}\right)\left(\genfrac{}{}{0}{}{k-mi}{i}\right)+\sum _{j=1}^{{q}_{\left(k\right)}-1}{B}^{0}{C}^{j}\left(\genfrac{}{}{0}{}{j-1}{0}\right)\left(\genfrac{}{}{0}{}{k-nj}{j}\right)\\ +\sum _{i=1,j=1}^{{p}_{\left(k\right)}-1,{q}_{\left(k\right)}-1}{B}^{i}{C}^{j}\left(\genfrac{}{}{0}{}{i+j-1}{i-1}\right)\left(\genfrac{}{}{0}{}{k-mi-nj}{i+j}\right)\\ +\sum _{i=1,j=1}^{{p}_{\left(k\right)}-1,{q}_{\left(k\right)}-1}{B}^{i}{C}^{j}\left(\genfrac{}{}{0}{}{i+j-1}{i}\right)\left(\genfrac{}{}{0}{}{k-mi-nj}{i+j}\right)+{C}^{{q}_{\left(k\right)}}\right]\\ =& \left(B+C\right)\left[I+\sum _{i=1,j=0}^{{p}_{\left(k\right)}-1,{q}_{\left(k\right)}-1}{B}^{i}{C}^{j}\left(\genfrac{}{}{0}{}{i+j-1}{i-1}\right)\left(\genfrac{}{}{0}{}{k-mi-nj}{i+j}\right)\\ +\sum _{i=0,j=1}^{{p}_{\left(k\right)}-1,{q}_{\left(k\right)}-1}{B}^{i}{C}^{j}\left(\genfrac{}{}{0}{}{i+j-1}{i}\right)\left(\genfrac{}{}{0}{}{k-mi-nj}{i+j}\right)+{C}^{{q}_{\left(k\right)}}\right].\end{array}$

Now we replace in the first sum the summation index i by $i+1$ and in the second sum we replace the summation index j by $j+1$. Then

$\begin{array}{rl}\mathrm{\Delta }{\mathrm{e}}_{mn}^{BCk}=& \left(B+C\right)\left[I+\sum _{i=0,j=0}^{{p}_{\left(k\right)}-2,{q}_{\left(k\right)}-1}{B}^{i+1}{C}^{j}\left(\genfrac{}{}{0}{}{i+j}{i}\right)\left(\genfrac{}{}{0}{}{k-m\left(i+1\right)-nj}{i+j+1}\right)\\ +\sum _{i=0,j=0}^{{p}_{\left(k\right)}-1,{q}_{\left(k\right)}-2}{B}^{i}{C}^{j+1}\left(\genfrac{}{}{0}{}{i+j}{i}\right)\left(\genfrac{}{}{0}{}{k-mi-n\left(j+1\right)}{i+j+1}\right)+{C}^{{q}_{\left(k\right)}}\right]\\ =& B+B\left(B+C\right)\sum _{i=0,j=0}^{{p}_{\left(k\right)}-2,{q}_{\left(k\right)}-1}{B}^{i}{C}^{j}\left(\genfrac{}{}{0}{}{i+j}{i}\right)\left(\genfrac{}{}{0}{}{k-m\left(i+1\right)-nj}{i+j+1}\right)\\ +C+C\left(B+C\right)\sum _{i=0,j=0}^{{p}_{\left(k\right)}-1,{q}_{\left(k\right)}-2}{B}^{i}{C}^{j}\left(\genfrac{}{}{0}{}{i+j}{i}\right)\left(\genfrac{}{}{0}{}{k-mi-n\left(j+1\right)}{i+j+1}\right)+{C}^{{q}_{\left(k\right)}}\left(B+C\right)\\ =& B\left[I+\left(B+C\right)\sum _{i=0,j=0}^{{p}_{\left(k\right)}-2,{q}_{\left(k\right)}-1}{B}^{i}{C}^{j}\left(\genfrac{}{}{0}{}{i+j}{i}\right)\left(\genfrac{}{}{0}{}{k-m\left(i+1\right)-nj}{i+j+1}\right)\right]\\ +C\left[I+\left(B+C\right)\left(\sum _{i=0,j=0}^{{p}_{\left(k\right)}-1,{q}_{\left(k\right)}-2}{B}^{i}{C}^{j}\left(\genfrac{}{}{0}{}{i+j}{i}\right)\left(\genfrac{}{}{0}{}{k-mi-n\left(j+1\right)}{i+j+1}\right)+{C}^{{q}_{\left(k\right)}-1}\right)\right].\end{array}$

For $k={q}_{\left(k\right)}\left(n+1\right)$, we have

${C}^{{q}_{\left(k\right)}-1}=\sum _{i=0}^{{p}_{\left(k\right)}-1}{B}^{i}{C}^{{q}_{\left(k\right)}-1}\left(\genfrac{}{}{0}{}{i+{q}_{\left(k\right)}-1}{i}\right)\left(\genfrac{}{}{0}{}{k-mi-n\left({q}_{\left(k\right)}-1+1\right)}{i+{q}_{\left(k\right)}-1+1}\right),$

where

Thus,

$\begin{array}{rl}\mathrm{\Delta }{\mathrm{e}}_{mn}^{BCk}=& B\left[I+\left(B+C\right)\sum _{i=0,j=0}^{{p}_{\left(k\right)}-2,{q}_{\left(k\right)}-1}{B}^{i}{C}^{j}\left(\genfrac{}{}{0}{}{i+j}{i}\right)\left(\genfrac{}{}{0}{}{k-m\left(i+1\right)-nj}{i+j+1}\right)\right]\\ +C\left[I+\left(B+C\right)\sum _{i=0,j=0}^{{p}_{\left(k\right)}-1,{q}_{\left(k\right)}-1}{B}^{i}{C}^{j}\left(\genfrac{}{}{0}{}{i+j}{i}\right)\left(\genfrac{}{}{0}{}{k-mi-n\left(j+1\right)}{i+j+1}\right)\right]\\ =& B{\mathrm{e}}_{mn}^{BC\left(k-m\right)}+C{\mathrm{e}}_{mn}^{BC\left(k-n\right)}\end{array}$

and formula (27) is proved.

### IV. $k={p}_{\left(k\right)}\left(m+1\right)\wedge k={q}_{\left(k\right)}\left(n+1\right)$

In this case, we have (see similar combinations in cases II and III)

${p}_{\left(k-m\right)}={p}_{\left(k\right)},\phantom{\rule{2em}{0ex}}{p}_{\left(k+1\right)}={p}_{\left(k\right)}+1$

and

${q}_{\left(k-n\right)}={q}_{\left(k\right)},\phantom{\rule{2em}{0ex}}{q}_{\left(k+1\right)}={q}_{\left(k\right)}+1.$

Then

$\begin{array}{r}{\mathrm{e}}_{mn}^{BCk}=I+\left(B+C\right)\sum _{i=0,j=0}^{{p}_{\left(k\right)}-1,{q}_{\left(k\right)}-1}{B}^{i}{C}^{j}\left(\genfrac{}{}{0}{}{i+j}{i}\right)\left(\genfrac{}{}{0}{}{k-mi-nj}{i+j+1}\right),\\ {\mathrm{e}}_{mn}^{BC\left(k+1\right)}=I+\left(B+C\right)\sum _{i=0,j=0}^{{p}_{\left(k\right)},{q}_{\left(k\right)}}{B}^{i}{C}^{j}\left(\genfrac{}{}{0}{}{i+j}{i}\right)\left(\genfrac{}{}{0}{}{k+1-mi-nj}{i+j+1}\right)\end{array}$

and

${\mathrm{e}}_{mn}^{BC\left(k-m\right)}=I+\left(B+C\right)\sum _{i=0,j=0}^{{p}_{\left(k\right)}-1,{q}_{\left(k-m\right)}-1}{B}^{i}{C}^{j}\left(\genfrac{}{}{0}{}{i+j}{i}\right)\left(\genfrac{}{}{0}{}{k-m-mi-nj}{i+j+1}\right),$
(28)
${\mathrm{e}}_{mn}^{BC\left(k-n\right)}=I+\left(B+C\right)\sum _{i=0,j=0}^{{p}_{\left(k-n\right)}-1,{q}_{\left(k\right)}-1}{B}^{i}{C}^{j}\left(\genfrac{}{}{0}{}{i+j}{i}\right)\left(\genfrac{}{}{0}{}{k-n-mi-nj}{i+j+1}\right).$
(29)

As before,

and

So, we can substitute ${q}_{\left(k\right)}$ for ${q}_{\left(k-m\right)}$ in (28) and ${p}_{\left(k\right)}$ for ${p}_{\left(k-n\right)}$ in (29) and

${\mathrm{e}}_{mn}^{BC\left(k-m\right)}=I+\left(B+C\right)\sum _{i=0,j=0}^{{p}_{\left(k\right)}-1,{q}_{\left(k\right)}-1}{B}^{i}{C}^{j}\left(\genfrac{}{}{0}{}{i+j}{i}\right)\left(\genfrac{}{}{0}{}{k-m\left(i+1\right)-nj}{i+j+1}\right),$
(30)
${\mathrm{e}}_{mn}^{BC\left(k-n\right)}=I+\left(B+C\right)\sum _{i=0,j=0}^{{p}_{\left(k\right)}-1,{q}_{\left(k\right)}-1}{B}^{i}{C}^{j}\left(\genfrac{}{}{0}{}{i+j}{i}\right)\left(\genfrac{}{}{0}{}{k-mi-n\left(j+1\right)}{i+j+1}\right).$
(31)

It is easy to see that, due to (3), formula (30) can also be used instead of (28) if ${q}_{\left(k-m\right)}<1$ and formula (31) can also be used instead of (29) if ${p}_{\left(k-n\right)}<1$.

Now it is possible to prove the formula

$\begin{array}{rl}\mathrm{\Delta }{\mathrm{e}}_{mn}^{BCk}=& B{\mathrm{e}}_{mn}^{BC\left(k-m\right)}+C{\mathrm{e}}_{mn}^{BC\left(k-n\right)}\\ =& B\left[I+\left(B+C\right)\sum _{i=0,j=0}^{{p}_{\left(k\right)}-1,{q}_{\left(k\right)}-1}{B}^{i}{C}^{j}\left(\genfrac{}{}{0}{}{i+j}{i}\right)\left(\genfrac{}{}{0}{}{k-m\left(i+1\right)-nj}{i+j+1}\right)\right]\\ +C\left[I+\left(B+C\right)\sum _{i=0,j=0}^{{p}_{\left(k\right)}-1,{q}_{\left(k\right)}-1}{B}^{i}{C}^{j}\left(\genfrac{}{}{0}{}{i+j}{i}\right)\left(\genfrac{}{}{0}{}{k-mi-n\left(j+1\right)}{i+j+1}\right)\right].\end{array}$
(32)

By definition, we get

$\begin{array}{rl}\mathrm{\Delta }{\mathrm{e}}_{mn}^{BCk}=& {\mathrm{e}}_{mn}^{BC\left(k+1\right)}-{\mathrm{e}}_{mn}^{BCk}\\ =& I+\left(B+C\right)\sum _{i=0,j=0}^{{p}_{\left(k\right)},{q}_{\left(k\right)}}{B}^{i}{C}^{j}\left(\genfrac{}{}{0}{}{i+j}{i}\right)\left(\genfrac{}{}{0}{}{k+1-mi-nj}{i+j+1}\right)\\ -I-\left(B+C\right)\sum _{i=0,j=0}^{{p}_{\left(k\right)}-1,{q}_{\left(k\right)}-1}{B}^{i}{C}^{j}\left(\genfrac{}{}{0}{}{i+j}{i}\right)\left(\genfrac{}{}{0}{}{k-mi-nj}{i+j+1}\right)\\ =& \left(B+C\right)\left[\sum _{i=0,j=0}^{{p}_{\left(k\right)}-1,{q}_{\left(k\right)}-1}{B}^{i}{C}^{j}\left(\genfrac{}{}{0}{}{i+j}{i}\right)\left[\left(\genfrac{}{}{0}{}{k+1-mi-nj}{i+j+1}\right)-\left(\genfrac{}{}{0}{}{k-mi-nj}{i+j+1}\right)\right]\\ +\sum _{j=0}^{{q}_{\left(k\right)}}{B}^{{p}_{\left(k\right)}}{C}^{j}\left(\genfrac{}{}{0}{}{{p}_{\left(k\right)}+j}{{p}_{\left(k\right)}}\right)\left(\genfrac{}{}{0}{}{k+1-m{p}_{\left(k\right)}-nj}{{p}_{\left(k\right)}+j+1}\right)\\ +\sum _{i=0}^{{p}_{\left(k\right)}}{B}^{i}{C}^{{q}_{\left(k\right)}}\left(\genfrac{}{}{0}{}{i+{q}_{\left(k\right)}}{i}\right)\left(\genfrac{}{}{0}{}{k+1-mi-n{q}_{\left(k\right)}}{i+{q}_{\left(k\right)}+1}\right)\right].\end{array}$

With the aid of equations $k={p}_{\left(k\right)}\left(m+1\right)$, $k={q}_{\left(k\right)}\left(n+1\right)$, we get

and

$\begin{array}{rl}\mathrm{\Delta }{\mathrm{e}}_{mn}^{BCk}=& \left(B+C\right)\left[\sum _{i=0,j=0}^{{p}_{\left(k\right)}-1,{q}_{\left(k\right)}-1}{B}^{i}{C}^{j}\left(\genfrac{}{}{0}{}{i+j}{i}\right)\left(\genfrac{}{}{0}{}{k-mi-nj}{i+j}\right)+{B}^{{p}_{\left(k\right)}}+{C}^{{q}_{\left(k\right)}}\right]\\ =& \left(B+C\right)\left[I+\sum _{i=1}^{{p}_{\left(k\right)}-1}{B}^{i}{C}^{0}\left(\genfrac{}{}{0}{}{i}{i}\right)\left(\genfrac{}{}{0}{}{k-mi}{i}\right)+\sum _{j=1}^{{q}_{\left(k\right)}-1}{B}^{0}{C}^{j}\left(\genfrac{}{}{0}{}{j}{0}\right)\left(\genfrac{}{}{0}{}{k-nj}{j}\right)\\ +\sum _{i=1,j=1}^{{p}_{\left(k\right)}-1,{q}_{\left(k\right)}-1}{B}^{i}{C}^{j}\left(\genfrac{}{}{0}{}{i+j}{i}\right)\left(\genfrac{}{}{0}{}{k-mi-nj}{i+j}\right)+{B}^{{p}_{\left(k\right)}}+{C}^{{q}_{\left(k\right)}}\right].\end{array}$

By (10), we have

$\begin{array}{rl}\mathrm{\Delta }{\mathrm{e}}_{mn}^{BCk}=& \left(B+C\right)\left[I+\sum _{i=1}^{{p}_{\left(k\right)}-1}{B}^{i}{C}^{0}\left(\genfrac{}{}{0}{}{i-1}{i-1}\right)\left(\genfrac{}{}{0}{}{k-mi}{i}\right)+\sum _{j=1}^{{q}_{\left(k\right)}-1}{B}^{0}{C}^{j}\left(\genfrac{}{}{0}{}{j-1}{0}\right)\left(\genfrac{}{}{0}{}{k-nj}{j}\right)\\ +\sum _{i=1,j=1}^{{p}_{\left(k\right)}-1,{q}_{\left(k\right)}-1}{B}^{i}{C}^{j}\left(\genfrac{}{}{0}{}{i+j-1}{i-1}\right)\left(\genfrac{}{}{0}{}{k-mi-nj}{i+j}\right)\\ +\sum _{i=1,j=1}^{{p}_{\left(k\right)}-1,{q}_{\left(k\right)}-1}{B}^{i}{C}^{j}\left(\genfrac{}{}{0}{}{i+j-1}{i}\right)\left(\genfrac{}{}{0}{}{k-mi-nj}{i+j}\right)+{B}^{{p}_{\left(k\right)}}+{C}^{{q}_{\left(k\right)}}\right]\\ =& \left(B+C\right)\left[I+\sum _{i=1,j=0}^{{p}_{\left(k\right)}-1,{q}_{\left(k\right)}-1}{B}^{i}{C}^{j}\left(\genfrac{}{}{0}{}{i+j-1}{i-1}\right)\left(\genfrac{}{}{0}{}{k-mi-nj}{i+j}\right)\\ +\sum _{i=0,j=1}^{{p}_{\left(k\right)}-1,{q}_{\left(k\right)}-1}{B}^{i}{C}^{j}\left(\genfrac{}{}{0}{}{i+j-1}{i}\right)\left(\genfrac{}{}{0}{}{k-mi-nj}{i+j}\right)+{B}^{{p}_{\left(k\right)}}+{C}^{{q}_{\left(k\right)}}\right].\end{array}$

We replace in the first sum the summation index i by $i+1$ and in the second sum we substitute the summation index j by $j+1$. Then

$\begin{array}{rl}\mathrm{\Delta }{\mathrm{e}}_{mn}^{BCk}=& \left(B+C\right)\left[I+\sum _{i=0,j=0}^{{p}_{\left(k\right)}-2,{q}_{\left(k\right)}-1}{B}^{i+1}{C}^{j}\left(\genfrac{}{}{0}{}{i+j}{i}\right)\left(\genfrac{}{}{0}{}{k-m\left(i+1\right)-nj}{i+j+1}\right)\\ +\sum _{i=0,j=0}^{{p}_{\left(k\right)}-1,{q}_{\left(k\right)}-2}{B}^{i}{C}^{j+1}\left(\genfrac{}{}{0}{}{i+j}{i}\right)\left(\genfrac{}{}{0}{}{k-mi-n\left(j+1\right)}{i+j+1}\right)+{B}^{{p}_{\left(k\right)}}+{C}^{{q}_{\left(k\right)}}\right]\\ =& B+B\left(B+C\right)\sum _{i=0,j=0}^{{p}_{\left(k\right)}-2,{q}_{\left(k\right)}-1}{B}^{i}{C}^{j}\left(\genfrac{}{}{0}{}{i+j}{i}\right)\left(\genfrac{}{}{0}{}{k-m\left(i+1\right)-nj}{i+j+1}\right)+{B}^{{p}_{\left(k\right)}}\left(B+C\right)\\ +C+C\left(B+C\right)\sum _{i=0,j=0}^{{p}_{\left(k\right)}-1,{q}_{\left(k\right)}-2}{B}^{i}{C}^{j}\left(\genfrac{}{}{0}{}{i+j}{i}\right)\left(\genfrac{}{}{0}{}{k-mi-n\left(j+1\right)}{i+j+1}\right)+{C}^{{q}_{\left(k\right)}}\left(B+C\right)\\ =& B\left[I+\left(B+C\right)\left(\sum _{i=0,j=0}^{{p}_{\left(k\right)}-2,{q}_{\left(k\right)}-1}{B}^{i}{C}^{j}\left(\genfrac{}{}{0}{}{i+j}{i}\right)\left(\genfrac{}{}{0}{}{k-m\left(i+1\right)-nj}{i+j+1}\right)+{B}^{{p}_{\left(k\right)}-1}\right)\right]\\ +C\left[I+\left(B+C\right)\left(\sum _{i=0,j=0}^{{p}_{\left(k\right)}-1,{q}_{\left(k\right)}-2}{B}^{i}{C}^{j}\left(\genfrac{}{}{0}{}{i+j}{i}\right)\left(\genfrac{}{}{0}{}{k-mi-n\left(j+1\right)}{i+j+1}\right)+{C}^{{q}_{\left(k\right)}-1}\right)\right].\end{array}$

Because $k={p}_{\left(k\right)}\left(m+1\right)={q}_{\left(k\right)}\left(n+1\right)$, we can express ${B}^{{p}_{\left(k\right)}-1}$ and ${C}^{{q}_{\left(k\right)}-1}$ in the form

$\begin{array}{r}{B}^{{p}_{\left(k\right)}-1}=\sum _{j=0}^{{q}_{\left(k\right)}-1}{B}^{{p}_{\left(k\right)}-1}{C}^{j}\left(\genfrac{}{}{0}{}{{p}_{\left(k\right)}-1+j}{{p}_{\left(k\right)}-1}\right)\left(\genfrac{}{}{0}{}{k-m\left({p}_{\left(k\right)}-1+1\right)-nj}{{p}_{\left(k\right)}-1+j+1}\right),\\ {C}^{{q}_{\left(k\right)}-1}=\sum _{i=0}^{{p}_{\left(k\right)}-1}{B}^{i}{C}^{{q}_{\left(k\right)}-1}\left(\genfrac{}{}{0}{}{i+{q}_{\left(k\right)}-1}{i}\right)\left(\genfrac{}{}{0}{}{k-mi-n\left({q}_{\left(k\right)}-1+1\right)}{i+{q}_{\left(k\right)}-1+1}\right),\end{array}$

where

Thus,

$\begin{array}{rl}\mathrm{\Delta }{\mathrm{e}}_{mn}^{BCk}=& B\left[I+\left(B+C\right)\sum _{i=0,j=0}^{{p}_{\left(k\right)}-1,{q}_{\left(k\right)}-1}{B}^{i}{C}^{j}\left(\genfrac{}{}{0}{}{i+j}{i}\right)\left(\genfrac{}{}{0}{}{k-m\left(i+1\right)-nj}{i+j+1}\right)\right]\\ +C\left[I+\left(B+C\right)\sum _{i=0,j=0}^{{p}_{\left(k\right)}-1,{q}_{\left(k\right)}-1}{B}^{i}{C}^{j}\left(\genfrac{}{}{0}{}{i+j}{i}\right)\left(\genfrac{}{}{0}{}{k-mi-n\left(j+1\right)}{i+j+1}\right)\right]\\ =& B{\mathrm{e}}_{mn}^{BC\left(k-m\right)}+C{\mathrm{e}}_{mn}^{BC\left(k-n\right)}.\end{array}$

Therefore, formula (32) is valid.

We proved that formula (6) holds in each of the considered cases I, II, III and IV for $k\ge 1$. If $k=0$, the proof can be done directly because ${p}_{\left(0\right)}={q}_{\left(0\right)}=0$, ${p}_{\left(1\right)}={q}_{\left(1\right)}=1$,

$\begin{array}{rl}\mathrm{\Delta }{\mathrm{e}}_{mn}^{BC0}=& {\mathrm{e}}_{mn}^{BC1}-{\mathrm{e}}_{mn}^{BC0}\\ =& I+\left(B+C\right)\sum _{i=0,j=0}^{0,0}{B}^{i}{C}^{j}\left(\genfrac{}{}{0}{}{i+j}{i}\right)\left(\genfrac{}{}{0}{}{1-mi-nj}{i+j+1}\right)\\ -I-\left(B+C\right)\sum _{i=0,j=0}^{-1,-1}{B}^{i}{C}^{j}\left(\genfrac{}{}{0}{}{i+j}{i}\right)\left(\genfrac{}{}{0}{}{-mi-nj}{i+j+1}\right)=I+B+C-I=B+C\end{array}$

and

$B{\mathrm{e}}_{mn}^{BC\left(-m\right)}+C{\mathrm{e}}_{mn}^{BC\left(-n\right)}=BI+CI=B+C.$

Formula (6) holds again. Theorem 2 is proved. □

## Open problems and concluding remarks

Formula (4) is valid for $k\in {\mathbb{Z}}_{-m}^{\mathrm{\infty }}$. However, formula (6) holds for $k\in {\mathbb{Z}}_{0}^{\mathrm{\infty }}$ only. Therefore, there is a difference between the definition domains of the formulas, and it is a challenge how to modify Definition 2 of discrete matrix delayed exponential for two delays in such a way that formula (6) will hold for $k\in {\mathbb{Z}}_{-max\left\{m,n\right\}}^{\mathrm{\infty }}$. In  formula (4) is used to get a representation of the solution of the problems (both homogeneous and nonhomogeneous)

$\begin{array}{r}\mathrm{\Delta }y\left(k\right)=By\left(k-m\right)+f\left(k\right),\phantom{\rule{1em}{0ex}}k\in {\mathbb{Z}}_{0}^{\mathrm{\infty }},\\ y\left(k\right)=\phi \left(k\right),\phantom{\rule{1em}{0ex}}k\in {\mathbb{Z}}_{-m}^{0},\end{array}$

where $f:{\mathbb{Z}}_{0}^{\mathrm{\infty }}\to {\mathbb{R}}^{r}$, $y:{\mathbb{Z}}_{-m}^{\mathrm{\infty }}\to {\mathbb{R}}^{r}$ and $\phi :{\mathbb{Z}}_{-m}^{0}\to {\mathbb{R}}^{r}$.

It is an open problem how to use formula (6) to get a representation of the solution of the homogeneous and nonhomogeneous problems

$\begin{array}{r}\mathrm{\Delta }y\left(k\right)=By\left(k-m\right)+Cy\left(k-n\right)+f\left(k\right),\phantom{\rule{1em}{0ex}}k\in {\mathbb{Z}}_{0}^{\mathrm{\infty }},\\ y\left(k\right)=\phi \left(k\right),\phantom{\rule{1em}{0ex}}k\in {\mathbb{Z}}_{-s}^{0},s=max\left\{m,n\right\}\end{array}$

if $BC=CB$.

Let us note that the first concept of matrix delayed exponential was given in  and the first concept of discrete matrix delayed exponential was given in . Further development of the delayed matrix exponentials method and its utilization to various problems can be found, e.g., in .

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## Acknowledgements

The first author was supported by Operational Programme Research and Development for Innovations, No. CZ.1.05/2.1.00/03.0097, as an activity of the regional Centre AdMaS.

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Correspondence to Josef Diblík.

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The authors declare that they have no competing interests.

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Diblík, J., Morávková, B. Discrete matrix delayed exponential for two delays and its property. Adv Differ Equ 2013, 139 (2013). https://doi.org/10.1186/1687-1847-2013-139

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• DOI: https://doi.org/10.1186/1687-1847-2013-139

### Keywords

• Main Property
• Constant Matrix
• Constant Matrice
• Binomial Coefficient
• Relevant Computation 