# Differential and Integral Calculus of Sequence

Yusuke Imai CONTACT<EMAIL_ADDRESS>Graduate School of Engineering
Science, Osaka University, Toyonaka, Osaka 560-8531, Japan

###### Abstract

We create a sequence version of calculus. First, we define equivalence, some
fundamental operations, differential, and integral for sequences. Then, we
propose sequence versions of identity function, power function, exponential
function, hyperbolic function, trigonometric function, and also find sequence
versions of the Maclaurin series for them. The sequence versions of
exponential function involve divergent series including Grandi’s series. By
using this framework, we find a sequence version of the binomial theorem and
Euler’s identity. In addition, we design new formalisms of Fibonacci sequence
and its generalizations. Last, we propose a sequence dual of factorial and
Bell number, and find sequence dual of modular property of factorial
concerning prime number (Wilson’s theorem) and of Bell number concerning prime
number.

###### Contents

  1. 1 Introduction
  2. 2 Definitions
    1. 2.1 Sequence
    2. 2.2 Equivalence
    3. 2.3 Fundamental operations
    4. 2.4 Differential and Integral
  3. 3 Fundamental sequence
  4. 4 Power function
  5. 5 Exponential function
    1. 5.1 $e^{x}$
    2. 5.2 $e^{-x}$
    3. 5.3 $e^{\alpha x}$
      1. 5.3.1 $\\{a_{n}[e^{\alpha x}]\\}_{R},\ \alpha\neq-1$
      2. 5.3.2 $\\{a_{n}[e^{\alpha x}]\\}_{L},\ \alpha\neq 1$
  6. 6 Hyperbolic function
  7. 7 Trigonometric function
  8. 8 Fibonacci sequences
    1. 8.1 Fibonacci sequence
    2. 8.2 $(P,\,Q)$-Fibonacci sequence
      1. 8.2.1 Example 1: Pell sequence ($P=2$, $Q=1$)
      2. 8.2.2 Example 2: Jacobsthal sequence ($P=1$, $Q=2$)
    3. 8.3 $k$-bonacci sequence
  9. 9 Factorial and Bell number
    1. 9.1 Factorial
    2. 9.2 Bell number
  10. 10 Discussion
  11. A List of sequence version of functions and sequence duals

##  1 Introduction

A sequence is an ordered list of numbers and is a fundamental mathematical
concept that has been investigated since ancient times. The following sequence
would be the most basic sequence.

$\displaystyle 1,1,1,1,1,1,\cdots.$ (1)

This sequence has the following three properties. First, all the numbers in
the above sequence are 1. Second, all the differences between the two
consecutive terms are 0. By this property, we can see that the above sequence
is an arithmetic sequence. Third, all the ratios between the two consecutive
terms are 1. By this property, we can see that the above sequence is a
geometric sequence. The following sequence is also a fundamental sequence that
is an arithmetic sequence whose common difference is 1 and the initial term is
given by 0.

$\displaystyle 0,1,2,3,4,5,\cdots.$ (2)

The following sequence is also a fundamental sequence that is a geometric
sequence whose common ratio is 2 and the initial term is given by 1.

$\displaystyle 1,2,4,8,16,32,\cdots.$ (3)

Each term of arithmetic sequences or geometric sequences is uniquely
determined by the previous term except the first term.

There exist sequences, each of whose term is determined by the previous two
terms, except the first two terms. The most famous one would be the sequence
each of whose term is given by sum of the previous two terms, except the first
two terms and that starts with 0 and 1, i.e. the Fibonacci sequence [1].

$\displaystyle 0,1,1,2,3,5,\cdots.$ (4)

The most characteristic feature of the Fibonacci sequence is that the ratio of
consecutive two terms in the Fibonacci sequence approach to the golden ratio.
The Fibonacci sequence was proposed in 1202 by Leonardo of Pisa [1], and it
has been discovered that it characterizes many things in nature, such as bee
populations, spiral patterns of poplar, willow, pear trees, beech, hazel,
cherry, apple, elm, lime, and almond [2].

In addition, there exist sequences, each of whose term is determined by its
index and the previous terms, except the first term. For example, each term of
the following sequence is given by $n$ times the previous term, except the
first term if the first index is set to 0.

$\displaystyle 1,1,2,6,24,120,\cdots.$ (5)

The $n$th term in the above sequence equals to factorial of $n$ that
represents the number of permutations for $n$ elements. The ($n$, $k$)th
unsigned Stirling numbers of the first kind represents the number of
permutations for $n$ elements with $k$ disjoint cyclic permutations. Then, sum
of the ($n$, $k$)th unsigned Stirling number of the first kind from $k=0$ to
$n$ equals to the $n$th factorial [3] (Fig. 1(a)).

The following sequence is called Bell numbers and is also a sequence that has
combinatorial meaning and can be considered as a dual of factorial in terms of
the Stirling numbers [4, 5, 6].

$\displaystyle 1,1,2,5,15,52,\cdots.$ (6)

The $n$th Bell number represents the number of ways to group $n$ elements. The
($n$, $k$)th Stirling number of the second kind represents the number of ways
to group $n$ elements by $k$ disjoint groups. Then, sum of the ($n$, $k$)th
Stirling number of the second kind from $k=0$ to $n$ equals to the $n$th Bell
number. While the ($n$, $k$)th unsigned Stirling numbers of the first kind can
also be defined as coefficients in a linear combination of $x^{k}$ for
$k=0,\cdots,n$ that equals to the rising factorial, $x(x+1)\cdots(x+n-1)$, the
($n$, $k$)th Stirling number of the second kind can also be defined as
coefficients in a linear combination of the falling factorial
$x(x-1)\cdots(x-k+1)$ for $k=0,\cdots,n$ that equals to $x^{n}$ [3] (Fig.
1(b)). Furthermore, both the factorials and the Bell numbers have the modular
property concerning prime numbers. The following theorem is called Wilson’s
theorem [7, 8, 9] and states that the $(p-1)$th factorial equals to $-1$
modulo $p$ if and only if $p$ is a prime number.

######  Theorem.

$\displaystyle(p-1)!\equiv-1\ ({\rm mod}\ p).$ (7)

Also, sum of the $n$th Bell number and $(n+1)$th Bell number equals to
$(n+p)$th Bell number modulo $p$ if $p$ is a prime number [10].

######  Theorem.

$\displaystyle B_{n+p}\equiv B_{n}+B_{n+1}\ ({\rm mod}\ p).$ (8)

Figure 1: Diagrammatic representation of (a) factorial and the unsigned
Stirling number of the first kind and (b) Bell number and Stirling number of
the second kind for $n=3$. The left diagram shows permutations of vertices of
a triangle for the arrangement of the bottom figure. The permutations are
represented by the circles, and the number of the circles corresponds to $k$.
The right diagram shows grouping of three numbers. The groups are represented
by the circles, and the number of the circles corresponds to $k$.

On the other hand, calculus is a fundamental mathematical branch that deals
with change or accumulation of a quantity [11]. The following real function
would be the most basic function.

$\displaystyle f(x)=1.$ (9)

This function has the following three properties. First, all the values the
above function takes are 1. Second, $f(x+1)-f(x)=0$ for all $x$ in real
numbers. Third, $f(x+1)/f(x)=1$ for all $x$ in real numbers. The following
real function is also a fundamental real function that is called identity
function and that satisfies $f(x+1)-f(x)=1$.

$\displaystyle f(x)=x.$ (10)

The following theorem is called binomial theorem [12].

######  Theorem.

$\displaystyle(1+x)^{n}=\sum_{k=0}^{n}{}_{n}C_{k}\,x^{k}.$ (11)

If $x=1$, the binomial theorem can be visualized by Pascal’s triangle (Fig.
2).

Figure 2: Pascal’s triangle.

The following real function is a fundamental real function that is called
exponential function and that satisfies $f(x+1)/f(x)=e$ where $e$ is the
Napier’s constant.

$\displaystyle f(x)=e^{x}.$ (12)

The exponential function is often used to represent an explosive increase of
quantity such as the number of infected persons [13]. The exponential function
is also characterized by invariance for differential as follows.

$\displaystyle\frac{d}{dx}e^{x}=e^{x}.$ (13)

The hyperbolic functions, $\sinh x$ and $\cosh x$, are determined by the
exponential function as follows.

$\displaystyle\cosh x$ $\displaystyle=\frac{e^{x}+e^{-x}}{2},$ (14)
$\displaystyle\sinh x$ $\displaystyle=\frac{e^{x}-e^{-x}}{2}.$ (15)

Perhaps the most famous application of hyperbolic functions is the four-
dimensional rotation in special relativity [14]. The hyperbolic functions
satisfy the following property for differential.

$\displaystyle\frac{d}{dx}\cosh x$ $\displaystyle=\sinh x,$ (16)
$\displaystyle\frac{d}{dx}\sinh x$ $\displaystyle=\cosh x.$ (17)

Also, the trigonometric functions, $\cos x$ and $\sin x$, are determined by
the exponential function as follows.

$\displaystyle\cos x$ $\displaystyle=\frac{e^{ix}+e^{-ix}}{2},$ (18)
$\displaystyle\sin x$ $\displaystyle=\frac{e^{ix}-e^{-ix}}{2i}.$ (19)

They are used to represent rotation in Euclidean space [15], and $\cos x$ and
$\sin x$ have the following periodic property.

$\displaystyle\cos(x+2\pi)$ $\displaystyle=\cos x,$ (20)
$\displaystyle\sin(x+2\pi)$ $\displaystyle=\sin x.$ (21)

Also, the trigonometric functions satisfy the following property for
differential.

$\displaystyle\frac{d}{dx}\cos x$ $\displaystyle=-\sin x,$ (22)
$\displaystyle\frac{d}{dx}\sin x$ $\displaystyle=\cos x.$ (23)

One can see that the differential rule for the trigonometric functions is
similar to one of the hyperbolic functions. The following relations also
indicate the similarity between the trigonometric functions and the hyperbolic
functions.

$\displaystyle\cos^{2}x+\sin^{2}x=1,$ (24)
$\displaystyle\cosh^{2}x-\sinh^{2}x=1,$ (25)
$\displaystyle\cos^{2}x-\sin^{2}x=\cos 2x,$ (26)
$\displaystyle\cosh^{2}x+\sinh^{2}x=\cosh 2x,$ (27) $\displaystyle 2\cos x\sin
x=\sin 2x,$ (28) $\displaystyle 2\cosh x\sinh x=\sinh 2x.$ (29)

The exponential function, the hyperbolic functions, and the trigonometric
functions have the Maclaurin series, i.e. can be represented by the series of
$1,x,x^{2},\cdots$ with coefficients.

$\displaystyle e^{x}$ $\displaystyle=\sum_{n=0}^{\infty}\frac{x^{n}}{n!},$
(30) $\displaystyle\cosh x$
$\displaystyle=\sum_{n=0}^{\infty}\frac{x^{2n}}{(2n)!},$ (31)
$\displaystyle\sinh x$
$\displaystyle=\sum_{n=0}^{\infty}\frac{x^{2n+1}}{(2n+1)!},$ (32)
$\displaystyle\cos x$
$\displaystyle=\sum_{n=0}^{\infty}\frac{(-1)^{n}x^{2n}}{(2n)!},$ (33)
$\displaystyle\sin x$
$\displaystyle=\sum_{n=0}^{\infty}\frac{(-1)^{n}x^{2n+1}}{(2n+1)!}.$ (34)

By differentiating the above Maclaurin series by term, one can show that they
satisfy the differential rules shown above.

Also, the exponential function relates to the trigonometric functions as
follows.

$\displaystyle e^{ix}=\cos x+i\sin x.$ (35)

The following equality is called Euler’s identity [16] and would be the most
fundamental and beautiful theorem in mathematics because it connects various
mathematical concepts, the $e$ (Napier’s constant), $0$ (additive identity),
$1$ (multiplicative identity), $i$ (imaginary unit), $\pi$ (circle ratio).

######  Theorem.

$\displaystyle e^{i\pi}+1=0.$ (36)

Figure 3(a) shows Euler’s identity ($e^{i\pi}+1=0$) and Fig. 3(b) shows
$e^{2i\pi}=1$, and Fig. 3(c) shows geometric representation of $e^{i\pi}\times
e^{i\pi}=1$.

Figure 3: Diagrammatic representation of (a) Euler’s identity ($e^{i\pi}+1=0$)
and (b) $e^{2i\pi}=1$, and (c) geometric representation of $e^{i\pi}\times
e^{i\pi}=1$ (area of the square equals to 1).

In addition, we introduce a function relating to the factorial and a function
relating to the Bell number. The following function is called gamma function
and can be defined in complex numbers whose real part is positive [17].

$\displaystyle\Gamma(z)=\int_{0}^{\infty}t^{z-1}e^{-t}dt.$ (37)

The gamma function satisfies the following relation

$\displaystyle\Gamma(n+1)=n!,$ (38)

i.e. the gamma function can be regarded as an extension of the factorial.

Also, the exponential function of the exponential function divided by $e$ is
represented as follows [4].

$\displaystyle e^{e^{x}}/e=\sum_{n=0}^{\infty}\frac{B_{n}x^{n}}{n!}.$ (39)

Then, the above function is called the exponential generating function of
$B_{n}$. Note that the exponential generating function of $1,1,1,\cdots$ is
given by the exponential function.

This paper investigates relations among the sequences and functions shown
above and their generalizations by creating a sequence version of calculus. In
Ref. [18], differential and integral of a sequence were proposed and they are
applied to solve discrete differential equations. In this paper, by inheriting
the concepts proposed in Ref. [18] with some retouching, we build a sequence
version of calculus. Because of the difference between the discreteness and
the continuity, it is difficult to construct a sequence version of calculus
that is perfectly analogous to usual calculus. However, we will see there
exist many beautiful sequence-analogies of calculus.

While writing this paper, we found a note [19] that tried to find a discrete
version of some elementary functions: exponential function, logarithmic
function, and trigonometric function. The note concludes with the sentences
after dealing with the discrete version of the trigonometric functions ($f(x)$
and $g(x)$); “Looking at this, the author wonders if $f(x)$ and $g(x)$ are
really the most appropriate discrete version of the trigonometric functions.
Could not a more appropriate function be found? What do the readers think?”.
Because, in this paper, we propose three kinds of sequence version of the
trigonometric functions and one of them corresponds to $f(x)$ and $g(x)$ (Eqs.
(179) and (180)) and the three sequence versions of the trigonometric
functions are entangled deeply each other to result in various formulae that
can be regarded as sequence version of formulae of the trigonometric
functions, we wish this paper can be one of the answers to that question.

In Sec. 2, we define some concepts relating to sequence. First, we show how to
represent a sequence and define equivalence between two sequences. Next, we
define fundamental operations including shift operator, sum, subtraction,
multiplication, division, scalar addition and multiplication, inverse, and
insertion. Then, we define two kinds of differential and integral: left
differential, right differential, left integral, right integral, and reveal
the relations among them.

In Sec. 3, we define the sequence version of fundamental real functions such
as additive identity ($f(x)=0$), multiplicative identity ($f(x)=1$), and
generally constant function ($f(x)=a$) where $a$ is a real number. We also
introduce the general notation that represents the sequence version of a real
function.

In Sec. 4, we define the sequence version of the identity function ($f(x)=x$),
and generally power function ($f(x)=x^{a}$) where $a$ is a real number. Also,
we show the law of exponent holds for the sequence version of the power
function. Then, we obtain a direct analogy of the binomial theorem (Eq. (11)).
This can be proved by using the usual binomial theorem.

In Sec. 5, we first define the sequence version of exponential function by
focusing on the property of the exponential function for differential (Eq.
(13)). Then, we obtain the sequence versions of the Maclaurin series for the
exponential function, some of which relates to divergent series including
Grandi’s series. Also, we find the right/left integral of the sequence version
of $x^{n}$ relates to $x^{n+1}$ through the Eulerian number. Next, we define
two kinds of sequence version of the negative exponential function depending
on whether the left differential or right differential is used. Last, we
define two kinds of sequence versions of the general exponential function
whose exponent is given by a general real number $a$.

In Sec. 6, we define two kinds of sequence version of the hyperbolic functions
by using the sequence version of the exponential function and the two kinds of
the negative exponential function defined in Sec. 5. Each of them satisfies
the sequence version of Eq. (25), Eq. (27), and Eq. (29).

In Sec. 7, we first define two kinds of sequence version of the sine function
and cosine function by using Eq. (35) and the two kinds of sequence version of
the general exponential function defined in Sec. 5. Each of them satisfies the
sequence version of Eq. (24), Eq. (26), and Eq. (28). Then, we obtain sequence
version of Euler’s identity (Eq. (36)). Also, by combining the two sequence
versions of the sine (cosine) function, we obtain another sequence version of
the sine (cosine) function whose periodicity is 8.

In Sec. 8, we propose the sequence version of the Maclaurin series for the
Fibonacci sequence, $(P,\,Q)$-Fibonnaci sequence including Pell number ($P=2$,
$Q=1$) and Jacobsthal number ($P=1$, $Q=2$), and $k$-bonacci sequence. We find
similarities between those generalizations of the Fibonacci sequence and the
sequence version of exponential function from the viewpoint of the sequence
version of the Maclaurin series.

In Sec. 9, we introduce sequence dual of factorial and of Bell numbers by
focusing on the functions shown in Eq. (37) and Eq. (39), and we find that
sequence dual of factorial (Bell numbers) is characterized by the Stirling
transform of the second (first) kind for the Bell numbers (factorial). In
addition, we find that the sequence dual of the factorial (Bell numbers)
inherits partially the modular property of the factorial (Bell numbers) for
the prime numbers shown in Eq. (7) (Eq. (8)).

Appendix A contains list of sequence version of functions and sequence duals
covered in this paper.

##  2 Definitions

In this section, we show how to represent a sequence and definition of
equivalence, sum, subtraction, multiplication, division, scalar addition and
multiplication, inverse, insertion, differential, and integral for sequence.

###  2.1 Sequence

In this paper, we represent a sequence that is ordered natural numbers, or
real numbers, or complex numbers, mapped from non-negative integers as
follows.

$\displaystyle\\{a_{n}\\}=a_{0},a_{1},a_{2},\cdots.$ (40)

###  2.2 Equivalence

First, we define the equivalence of two sequences as follows.

######  Definition 1 (Equivalence).

For any sequences $\\{a_{n}\\}$ and $\\{b_{n}\\}$, $\\{a_{n}\\}=\\{b_{n}\\}$
if and only if for any non-negative integer $n$,

$\displaystyle a_{n}=b_{n}.$ (41)

###  2.3 Fundamental operations

In this subsection, we define fundamental operations for a sequence or two
sequences. First, we define shift operator $\mathcal{S}_{k}$ as follows.

######  Definition 2 (Shift operator).

For any sequence $\\{a_{n}\\}$ and non-negative integer $k$,

$\displaystyle\mathcal{S}_{k}a_{n}=a_{n+k},\ \
\mathcal{S}_{k}\\{a_{n}\\}=\\{a_{n+k}\\}.$ (42)

The sum ($+$), subtraction ($-$), multiplication ($\times$), and division
($/$) of two sequences are naturally defined as follows.

######  Definition 3 (Sum, Subtraction, Multiplication, Division).

For any sequences $\\{a_{n}\\}$ and $\\{b_{n}\\}$,

$\displaystyle\\{a_{n}\\}+\\{b_{n}\\}$ $\displaystyle=\\{a_{n}+b_{n}\\},$ (43)
$\displaystyle\\{a_{n}\\}-\\{b_{n}\\}$ $\displaystyle=\\{a_{n}-b_{n}\\},$ (44)
$\displaystyle\\{a_{n}\\}\times\\{b_{n}\\}$ $\displaystyle=\\{a_{n}\times
b_{n}\\}.$ (45)

For any sequence $\\{a_{n}\\}$ and $\\{b_{n}\\}$ that does not include $0$,

$\displaystyle\\{a_{n}\\}/\\{b_{n}\\}=\\{a_{n}/b_{n}\\}.$ (46)

Scalar addition and scalar multiplication are also naturally defined as
follows.

######  Definition 4 (Scalar addition and multiplication).

For any sequence $\\{a_{n}\\}$ and a real number $\alpha$,

$\displaystyle\\{a_{n}+\alpha\\}$ $\displaystyle=\\{a_{n}\\}+\alpha,$ (47)
$\displaystyle\\{\alpha a_{n}\\}$ $\displaystyle=\alpha\\{a_{n}\\}.$ (48)

Inverse of a given sequence $\\{a_{n}\\}$ that does not include 0 is defined
as follows.

######  Definition 5 (Inverse).

We represent inverse of $\\{a_{n}\\}$ that does not include 0 by
$\\{a_{n}\\}^{-1}$ and define $\\{a_{n}\\}^{-1}$ as

$\displaystyle\\{a_{n}\\}^{-1}=a_{0}^{-1},\ a_{1}^{-1},\cdots.$ (49)

Also, we define insertion operator $I_{\alpha}$ for a real number $\alpha$ as
follows.

######  Definition 6 (Insertion).

$\displaystyle I_{\alpha}\\{a_{n}\\}=\alpha,a_{0},a_{1},\cdots.$ (50)

Note that the insertion operator is relate to the shift operator as follows.

$\displaystyle\mathcal{S}_{1}I_{\alpha}\\{a_{n}\\}=\\{a_{n}\\}.$ (51)

###  2.4 Differential and Integral

In this subsection, we define differential and integral of sequence. First, we
propose two definitions of the differential of sequence: right differential
and left differential. The right differential $\mathcal{D}_{R}\\{a_{n}\\}_{R}$
of a sequence $\\{a_{n}\\}$ is defined as follows.

######  Definition 7 (Right Differential).

For any sequence $\\{a_{n}\\}$,

$\displaystyle\mathcal{D}_{R}\\{a_{n}\\}=\\{a_{n+1}-a_{n}\\}.$ (52)

We also use the left differential of $\mathcal{D}_{L}\\{a_{n}\\}_{L}$ of a
sequence $\\{a_{n}\\}$.

######  Definition 8 (Left differential).

For any sequence $\\{a_{n}\\}$,

$\displaystyle\mathcal{D}_{L}\\{a_{n}\\}=\\{a_{n}-a_{n-1}\\}.$ (53)

To make the above definition well defined, $a_{-1}$ is needed. We add an
appropriate $a_{-1}$ to $\\{a_{n}\\}$ if necessary.

We remark that the following relation of the right differential and the left
differential holds.

$\displaystyle\mathcal{D}_{R}=\mathcal{S}_{1}\mathcal{D}_{L}.$ (54)

We also define the right integral $\mathcal{I}_{R}^{\alpha}\\{a_{n}\\}$ of a
sequence $\\{a_{n}\\}$ is defined as follows.

######  Definition 9 (Right Integral).

For any sequence $\\{a_{n}\\}$ and an integral constant that is a real number,
$\alpha$,

$\displaystyle\mathcal{I}_{R}^{\alpha}\\{a_{n}\\}=\left\\{\sum_{k=0}^{n}a_{k}\right\\}+\alpha.$
(55)

In addition, we define the left integral $\mathcal{I}_{L}^{\alpha}\\{a_{n}\\}$
of a sequence $\\{a_{n}\\}$ as follows.

######  Definition 10 (Left Integral).

For any sequences $\\{a_{n}\\}$ and an integral constant that is a real
number, $\alpha$,

$\displaystyle\mathcal{I}_{L}^{\alpha}\\{a_{n}\\}=\left\\{\sum_{k=0}^{n-1}a_{k}\right\\}+\alpha,$
(56)

where we assumed $\sum_{k=0}^{-1}a_{k}=0$.

We remark that the following relation of the right differential and the left
differential holds.

$\displaystyle\mathcal{I}_{R}^{\alpha}=\mathcal{S}_{1}\mathcal{I}_{L}^{\alpha}.$
(57)

One can check that the right/left integral is the inverse operation of the
left/right differential as follows.

$\displaystyle\mathcal{D}_{L}\mathcal{I}_{R}^{\alpha}\\{a_{n}\\}$
$\displaystyle=\left\\{\sum_{k=0}^{n}a_{k}-\sum_{k=0}^{n-1}a_{k}\right\\}=\\{a_{n}\\},$
(58) $\displaystyle\mathcal{I}_{R}^{\alpha}\mathcal{D}_{L}\\{a_{n}\\}$
$\displaystyle=\left\\{\sum_{k=0}^{n}(a_{k}-a_{k-1})\right\\}+\alpha=\\{a_{n}\\}-a_{-1}+\alpha,$
(59) $\displaystyle\mathcal{D}_{R}\mathcal{I}_{L}^{\alpha}\\{a_{n}\\}$
$\displaystyle=\left\\{\sum_{k=1}^{n}a_{k}-\sum_{k=1}^{n-1}a_{k}\right\\}=\\{a_{n}\\},$
(60) $\displaystyle\mathcal{I}_{L}^{\alpha}\mathcal{D}_{R}\\{a_{n}\\}$
$\displaystyle=\left\\{\sum_{k=0}^{n-1}(a_{k+1}-a_{k})\right\\}+\alpha=\\{a_{n}-a_{0}\\}+\alpha=\\{a_{n}\\}-a_{0}+\alpha.$
(61)

There exist the relations among shift operator, the left/right integral, and
the left/right differential as follows.

$\displaystyle\mathcal{D}_{R}\mathcal{I}_{R}^{\alpha}\\{a_{n}\\}$
$\displaystyle=\left\\{\sum_{k=0}^{n+1}a_{k}-\sum_{k=0}^{n}a_{k}\right\\}=\\{a_{n+1}\\}=\mathcal{S}_{1}\\{a_{n}\\},$
(62) $\displaystyle\mathcal{I}_{R}^{\alpha}\mathcal{D}_{R}\\{a_{n}\\}$
$\displaystyle=\left\\{\sum_{k=0}^{n}(a_{k+1}-a_{k})\right\\}=\\{a_{n+1}-a_{0}\\}+\alpha=\mathcal{S}_{1}\\{a_{n}\\}-a_{0}+\alpha,$
(63)
$\displaystyle\mathcal{S}_{1}\mathcal{D}_{L}\mathcal{I}_{L}^{\alpha}\\{a_{n}\\}$
$\displaystyle=\left\\{\sum_{k=0}^{n}a_{k}-\sum_{k=0}^{n-1}a_{k}\right\\}=\\{a_{n}\\},$
(64)
$\displaystyle\mathcal{S}_{1}\mathcal{I}_{L}^{\alpha}\mathcal{D}_{L}\\{a_{n}\\}$
$\displaystyle=\left\\{\sum_{k=0}^{n}(a_{k}-a_{k-1})\right\\}=\\{a_{n}-a_{0}\\}+\alpha=\\{a_{n}\\}-a_{0}+\alpha.$
(65)

In Fig. 4, we show how the left/right differential and integral act on a
sequence, $\\{a_{n}\\}=1,3,2,5,6,4,\cdots$ and $a_{-1}=0$.

Figure 4: The left/right differential and integral of a sequence
$\\{a_{n}\\}=1,3,2,5,6,4,\cdots$ and $a_{-1}=0$.

##  3 Fundamental sequence

In this section, we define two kinds of identity sequence: additive identity
sequence and multiplicative identity sequence, and sequence version of
constant function. Remembering the additive operation defined in the previous
section, the additive identity sequence $\\{a_{n}[0]\\}$ should be defined as
follows.

######  Definition 11 (Sequence version of 0).

$\displaystyle\\{a_{n}[0]\\}=\\{0\\}=0,\ 0,\ 0,\cdots.$ (66)

Then, any sequence $\\{a_{n}\\}$ satisfies the relations

$\displaystyle\\{a_{n}\\}+\\{a_{n}[0]\\}=\\{a_{n}[0]\\}+\\{a_{n}\\}=\\{a_{n}\\},$
(67) $\displaystyle\\{a_{n}\\}-\\{a_{n}[0]\\}=\\{a_{n}\\},$ (68)
$\displaystyle\\{a_{n}[0]\\}-\\{a_{n}\\}=\\{-a_{n}\\}=-\\{a_{n}\\}.$ (69)

In addition, the multiplicative identity sequence $\\{a_{n}[1]\\}$ should be
defined as follows.

######  Definition 12 (Sequence version of 1).

$\displaystyle\\{a_{n}[1]\\}=\\{1\\}=1,\ 1,\ 1,\cdots.$ (70)

Then, for any sequence $\\{a_{n}\\}$,

$\displaystyle\\{a_{n}\\}\times\\{a_{n}[1]\\}=\\{a_{n}[1]\\}\times\\{a_{n}\\}=\\{a_{n}\\},$
(71) $\displaystyle\\{a_{n}\\}/\\{a_{n}[1]\\}=\\{a_{n}\\},$ (72)

and for any sequence $\\{a_{n}\\}$ that does not include 0,

$\displaystyle\\{a_{n}[1]\\}/\\{a_{n}\\}=\\{a_{n}\\}^{-1}.$ (73)

Generally, sequence version of constant function $\\{a_{n}[a]\\}$ where $a$ is
a real number is given by

######  Definition 13 (Sequence version of $a$).

$\displaystyle\\{a_{n}[a]\\}=\\{a\\}=a,a,a,\cdots.$ (74)

The sequence version of constant function $\\{a_{n}[a]\\}$ satisfies the
following relations for any sequence $\\{a_{n}\\}$.

$\displaystyle\\{a_{n}\\}+\\{a_{n}[a]\\}=\\{a_{n}[a]\\}+\\{a_{n}\\}=\\{a_{n}+a\\}=\\{a_{n}\\}+a,$
(75) $\displaystyle\\{a_{n}\\}-\\{a_{n}[a]\\}=\\{a_{n}-a\\}=\\{a_{n}\\}-a,$
(76) $\displaystyle\\{a_{n}[a]\\}-\\{a_{n}\\}=\\{a-a_{n}\\}=a-\\{a_{n}\\},$
(77)
$\displaystyle\\{a_{n}\\}\times\\{a_{n}[a]\\}=\\{a_{n}[a]\\}\times\\{a_{n}\\}=\\{aa_{n}\\}=a\\{a_{n}\\},$
(78) $\displaystyle\\{a_{n}\\}/\\{a_{n}[a]\\}=\\{a_{n}/a\\}=\\{a_{n}\\}/a,$
(79)
$\displaystyle\\{a_{n}[a]\\}/\\{a_{n}\\}=\\{a/a_{n}\\}=a\\{a_{n}\\}^{-1},$
(80)

where we assumed $a\neq 0$ in Eq. (79) and $a_{n}\neq 0$ for any non-negative
integer $n$ in Eq. (80). In this paper, we use the following notation to
represent a sequence version of a given function $f(x)$,

######  Definition 14.

$\displaystyle\\{a_{n}[f(x)]\\}=a_{0}[f(x)],a_{1}[f(x)],\cdots.$ (81)

##  4 Power function

In this section, we define the sequence version of power function. First, we
define $\\{a_{n}\\}[x]$ that is the sequence version of $x$ as follows.

######  Definition 15 (Sequence version of $x$).

$\displaystyle\\{a_{n}[x]\\}=\\{n\\}=0,\ 1,\ 2,\cdots.$ (82)

Generally, we define the sequence version of power function as follows.

######  Definition 16 (Sequence version of $x^{a}$).

For a real number, $a$,

$\displaystyle\\{a_{n}[x^{a}]\\}=\\{n^{a}\\}=0^{a},\ 1^{a},\ 2^{a},\cdots.$
(83)

Note that for real numbers $a$, $b$, the following formulae are analogous to
the law of exponent.

######  Proposition 1 (Sequence version of the law of exponent).

$\displaystyle\\{a_{n}[x^{a+b}]\\}=\\{n^{a+b}\\}=0^{a+b},\ 1^{a+b},\
2^{a+b},\cdots$ $\displaystyle\qquad=0^{a}\times 0^{b},\ 1^{a}\times 1^{b},\
2^{a}\times 2^{b},\cdots=\\{a_{n}[x^{a}]\\}\times\\{a_{n}[x^{b}]\\},$ (84)
$\displaystyle\\{a_{n}[x^{a-b}]\\}=\\{n^{a-b}\\}=0^{a-b},\ 1^{a-b},\
2^{a-b},\cdots$ $\displaystyle\qquad=\frac{0^{a}}{0^{b}},\
\frac{1^{a}}{1^{b}},\
\frac{2^{a}}{2^{b}},\cdots=\\{a_{n}[x^{a}]\\}/\\{a_{n}[x^{b}]\\},$ (85)
$\displaystyle\\{a_{n}[x^{ab}]\\}=\\{n^{ab}\\}=0^{ab},1^{ab},2^{ab},\cdots$
$\displaystyle\qquad=(0^{a})^{b},(1^{a})^{b},(2^{a})^{b},\cdots=\\{a_{n}[x^{a}]\\}^{b}.$
(86)

By using power function, one has the following theorem called Binomial
theorem.

$\displaystyle(1+x)^{n}=\sum_{k=0}^{n}{}_{n}C_{k}x^{k}.$ (87)

Then, we find the sequence version of the binomial theorem as follows (Fig.
5).

######  Theorem 1 (Sequence version of binomial theorem).

For a non-negative number $n$,

$\displaystyle(\\{a_{n}[1]\\}+\\{a_{n}[x]\\})^{n}$
$\displaystyle\qquad={}_{n}C_{0}\\{a_{n}[1]\\}^{n}\\{a_{n}[x]\\}^{0}+{}_{n}C_{1}\\{a_{n}[1]\\}^{n-1}\\{a_{n}[x]\\}^{1}+\cdots+{}_{n}C_{n}\\{a_{n}[1]\\}^{0}\\{a_{n}[x]\\}^{n}$
(88)
$\displaystyle\qquad=\sum_{k=0}^{n}{}_{n}C_{k}\\{a_{n}[1]\\}^{n-k}\\{a_{n}[x]\\}^{k}.$
(89)

Figure 5: Diagrammatic representation of sequence version of binomial theorem.

##  5 Exponential function

In this section, we define the sequence versions of exponential function by
focusing on differential property of the exponential function.

###  5.1 $e^{x}$

The exponential function satisfies the following formula for differential.

$\displaystyle\frac{d}{dx}e^{x}=e^{x}.$ (90)

Then, we find a sequence $\\{a_{n}[e^{x}]\\}_{R}$ that satisfies the following
formula111In Remark 2, we define $\\{a_{n}[e^{x}]\\}_{L}$ by using
$\mathcal{D}_{L}$.

$\displaystyle\mathcal{D}_{R}\\{a_{n}[e^{x}]\\}_{R}=\\{a_{n}[e^{x}]\\}_{R}.$
(91)

The $\\{a_{n}[e^{x}]\\}_{R}$ can be uniquely determined except the initial
term as follows.

$\displaystyle a_{n+1}-a_{n}=a_{n},$ (92) $\displaystyle a_{n+1}=2a_{n}.$ (93)

Then, we define $\\{a_{n}[e^{x}]\\}_{R}$ as follows because $e^{0}=1$.

######  Definition 17 (Right sequence version of exponential function).

$\displaystyle\\{a_{n}[e^{x}]\\}_{R}=\\{2^{n}\\}=1,\ 2,\ 4,\cdots.$ (94)

Figure 6 shows difference between the usual exponential function ($e^{x}$) and
the right sequence version of the exponential function
($\\{a_{n}[e^{x}]\\}_{R}$).

Figure 6: Difference between $e^{x}$ and $\\{a_{n}[e^{x}]\\}_{R}$.

We remark that the difference between $d/dx$ and $\mathcal{D}_{R}$ is
manifested by the difference between the $e$ and the $2$. Then, we show
sequence version of the Maclaurin series for $e^{x}$. By using the Maclaurin
series, the $e^{x}$ is expressed as follows.

$\displaystyle e^{x}$
$\displaystyle=\sum_{n=0}^{\infty}\frac{x^{n}}{n!}=\frac{1}{0!}1+\frac{1}{1!}x+\frac{1}{2!}x^{2}+\frac{1}{3!}x^{3}+\cdots$
$\displaystyle=1+x+\int_{0}^{x}x+\int_{0}^{x}\int_{0}^{x}x+\int_{0}^{x}\int_{0}^{x}\int_{0}^{x}x+\cdots.$
(95)

Because we have already defined the sequence version of $1$, $x$, $e^{x}$, and
the left integral of a sequence, then, we have the following left sequence
version of the Maclaurin series for $\\{a_{n}[e^{x}]\\}_{R}$ as follows.

######  Proposition 2 (Left sequence version of Maclaurin series for right
sequence version of exponential function 1).

$\displaystyle\\{a_{n}[e^{x}]\\}_{R}=\\{a_{n}[1]\\}+\\{a_{n}[x]\\}+\mathcal{I}_{L}^{0}\\{a_{n}[x]\\}+\mathcal{I}_{L}^{0}\mathcal{I}_{L}^{0}\\{a_{n}[x]\\}+\mathcal{I}_{L}^{0}\mathcal{I}_{L}^{0}\mathcal{I}_{L}^{0}\\{a_{n}[x]\\}+\cdots.$
(96)

Each term in the r.h.s. in the above equation is given by

$\displaystyle\mathcal{I}_{L}^{0}\\{a_{n}[x]\\}$
$\displaystyle=0,0,1,3,6,10,\cdots,$ (97)
$\displaystyle\mathcal{I}_{L}^{0}\mathcal{I}_{L}^{0}\\{a_{n}[x]\\}$
$\displaystyle=0,0,0,1,4,10,20,\cdots,$ (98)
$\displaystyle\mathcal{I}_{L}^{0}\mathcal{I}_{L}^{0}\mathcal{I}_{L}^{0}\\{a_{n}[x]\\}$
$\displaystyle=0,0,0,0,1,5,15,35,\cdots,$ (99)

etc. Then, we introduce the following definition.

######  Definition 18 (Left sequence version of $x^{n}/n!$).

For an integer $k\geq 2$,

$\displaystyle\\{a_{n}[x^{k}/k!]\\}_{L}=\underbrace{\mathcal{I}_{L}^{0}\cdots\mathcal{I}_{L}^{0}}_{\rm
k-1\ times}\\{a_{n}[x]\\}.$ (100)

We remark
$\\{a_{n}[x^{2}]\\}\neq\\{a_{n}[x^{2}/2]\\}_{L}+\\{a_{n}[x^{2}/2]\\}_{L}$.
Instead, the following formula holds.

$\displaystyle\\{a_{n}[x^{2}]\\}=\\{a_{n}[x^{2}/2]\\}_{L}+\mathcal{S}_{1}\\{a_{n}[x^{2}/2]\\}_{L}.$
(101)

Generally, we obtain the following proposition.

######  Proposition 3 (Relation between left sequence version of $x^{n}/n!$
and sequence version of $x^{n}$).

For a natural number $n$,

$\displaystyle\\{a_{n}[x^{n}]\\}=\sum_{k=0}^{n-1}A(n,k)\mathcal{S}_{k}\\{a_{n}[x^{n}/n!]\\}_{L},$
(102)

where $A(n,k)$ is the Eulerian number satisfying the following formula for
$n=1,2,3,\cdots$ and $k=0,1,2,\cdots,n$ [20, 21, 22].

$\displaystyle A(n,k)=\sum_{l=0}^{k+1}(-1)^{l}\begin{pmatrix}n+1\\\
l\end{pmatrix}(k+1-l)^{n}.$ (103)

The Eulerian number represents the number of permutations for $n$ elements
with $k$ raises. For example, $A(3,2)=1$ because there exist only one
permutation for 3 elements with 2 raises: $(1,2,3)$. Note that $3-2=+1$ and
$2-1=+1$.

Then, we have the following formula.

######  Proposition 4 (Left sequence version of Maclaurin series for right
sequence version of exponential function 2).

$\displaystyle\\{a_{n}[e^{x}]\\}_{R}=\sum_{n=0}^{\infty}\\{a_{n}[x^{n}/n!]\\}_{L}.$
(104)

By using the right integral of a sequence, we have the following right
sequence version of the Maclaurin series for $\\{a_{n}[e^{x}]\\}_{R}$ as
follows.

######  Proposition 5 (Right sequence version of Maclaurin series for right
sequence version of exponential function 1).

$\displaystyle\\{a_{n}[e^{x}]\\}_{R}$
$\displaystyle=\\{a_{n}[1]\\}+\frac{1}{2}\\{a_{n}[x]\\}+\frac{1}{4}\mathcal{I}_{R}^{0}\\{a_{n}[x]\\}$
$\displaystyle\qquad+\frac{1}{8}\mathcal{I}_{R}^{0}\mathcal{I}_{R}^{0}\\{a_{n}[x]\\}+\frac{1}{16}\mathcal{I}_{R}^{0}\mathcal{I}_{R}^{0}\mathcal{I}_{R}^{0}\\{a_{n}[x]\\}+\cdots.$
(105)

Also, we have

######  Proposition 6 (Right sequence version of Maclaurin series for right
sequencne version of exponential function 2).

$\displaystyle\\{a_{n}[e^{x}]\\}_{R}=\sum_{n=0}^{\infty}\frac{1}{2^{n}}\\{a_{n}[x^{n}/n!]\\}_{R}.$
(106)

Here, we defined $\\{a_{n}[x^{n}/n!]\\}_{R}$ as follows.

######  Definition 19 (Right sequence version of $x^{n}/n!$).

For an integer $k\geq 2$,

$\displaystyle\\{a_{n}[x^{k}/k!]\\}_{R}=\underbrace{\mathcal{I}_{R}^{0}\cdots\mathcal{I}_{R}^{0}}_{\rm
k-1\ times}\\{a_{n}[x]\\}=\mathcal{S}_{k-1}\\{a_{n}[x^{k}/k!]\\}_{L}.$ (107)

For example, we have

$\displaystyle\mathcal{I}_{R}^{0}\\{a_{n}[x]\\}$
$\displaystyle=0,1,3,6,10,\cdots,$ (108)
$\displaystyle\mathcal{I}_{R}^{0}\mathcal{I}_{R}^{0}\\{a_{n}[x]\\}$
$\displaystyle=0,1,4,10,20,\cdots,$ (109)
$\displaystyle\mathcal{I}_{R}^{0}\mathcal{I}_{R}^{0}\mathcal{I}_{R}^{0}\\{a_{n}[x]\\}$
$\displaystyle=0,1,5,15,35,\cdots.$ (110)

Also, the first three terms of the right sequence version of the Maclaurin
series for $\\{a_{n}[e^{x}]\\}_{R}$ are represented as follows.

$\displaystyle 1+0+0+\cdots=1,$ (111) $\displaystyle
1+\frac{1}{2}+\frac{1}{4}+\cdots=\sum_{n=0}^{\infty}\frac{1}{2^{n}}=2,$ (112)
$\displaystyle
1+\frac{2}{2}+\frac{3}{4}+\cdots=\sum_{n=0}^{\infty}\frac{n+1}{2^{n}}=4.$
(113)

In addition, we obtain the following relation between the right sequence
version of $x^{n}/n!$ and sequence version of $x^{n}$.

######  Proposition 7 (Relation between right sequence version of $x^{n}/n!$
and sequence version of $x^{n}$).

For a natural number $n$,

$\displaystyle\\{a_{n}[x^{n}]\\}=\sum_{k=0}^{n-1}A(n,k)I_{0}^{n-1-k}\\{a_{n}[x^{n}/n!]\\}_{R},$
(114)

Note that the factors $1/2$, $1/2^{2}$ etc. are needed to construct the right
sequence version of Maclaurin series for $\\{a_{n}[e^{x}]\\}_{R}$, and
replacement of the $\int_{0}^{x}$ by the $\mathcal{I}_{L}^{0}$ does not
directly lead to the right sequence version of Maclaurin series for
$\\{a_{n}[e^{x}]\\}_{R}$. In this paper, the factors $1/2$, $1/2^{2}$ etc.
frequently play the role of bridge between the world of sequence and the world
of analytics.

We remark the following three points.

######  Remark 1.

We show two kinds of diagrams corresponding to Prop. 2 in Figs. 7 and 8.

Figure 7: Diagrammatic representation of left sequence version of Maclaurin
series for $\\{a_{n}[e^{x}]\\}_{R}$ that is equivalent to Pascal’s triangle.
Figure 8: Diagrammatic representation of left sequence version of Maclaurin
series for $\\{a_{n}[e^{x}]\\}_{R}$ by using blocks.

######  Remark 2.

We used $\mathcal{D}_{R}$ to define $\\{a_{n}[e^{x}]\\}_{R}$ in Eq. (91). The
$\\{a_{n}[e^{x}]\\}_{L}$ should satisfy the following properties for any non-
negative integer $n$.

$\displaystyle a_{n}-a_{n-1}$ $\displaystyle=a_{n},$ (115) $\displaystyle
a_{n-1}$ $\displaystyle=0.$ (116)

Then,

$\displaystyle\\{a_{n}[e^{x}]\\}_{L}=0,0,\cdots.$ (117)

######  Remark 3.

In Fig. 9, Prop. 3 for $n=3$ is represented.

Figure 9: Diagrammatic representation of the relation among
$\\{a_{n}[x^{n}]\\}$, $\\{a_{n}[x^{n}/n!]\\}_{L}$, and
$\\{a_{n}[x^{n}/n!]\\}_{R}$ with the Eulerian number $A(n,k)$ for $n=3$.

###  5.2 $e^{-x}$

Next, we define the sequence version of $e^{-x}$. The $e^{-x}$ satisfies the
following formula.

$\displaystyle\frac{d}{dx}e^{-x}=-e^{-x}.$ (118)

Then, we find a sequence $\\{a_{n}[e^{-x}]\\}_{L}$ that satisfies the
following formula.

$\displaystyle\mathcal{D}_{L}\\{a_{n}[e^{x}]\\}_{L}=-\\{a_{n}[e^{x}]\\}_{L}.$
(119)

The $\\{a_{n}[e^{-x}]\\}_{L}$ can uniquely be determined except the initial
term as follows.

$\displaystyle a_{n}-a_{n-1}=-a_{n},$ (120) $\displaystyle 2a_{n}=a_{n-1}.$
(121)

Then, we define
$\\{a_{n}[e^{-x}]\\}_{L}=a_{0}[e^{-x}]_{L},a_{1}[e^{-x}]_{L},\cdots$ as
follows because $e^{-0}=1$.

######  Definition 20 (Left sequence version of negative exponential
function).

$\displaystyle\\{a_{n}[e^{-x}]\\}_{L}=\\{2^{-n}\\}=1,\ \frac{1}{2},\
\frac{1}{4},\cdots,$ (122)

where we assumed $a_{-1}[e^{-x}]_{L}=2$.

By combining the right sequence version of exponential function and the left
sequence version of negative exponential function, we have the following
inverse relation.

######  Proposition 8 (Inverse relation between right sequence version of
exponential function and left sequence version of negative exponential
function).

$\displaystyle\\{a_{n}[(e^{x})]\\}_{R}^{-1}=\\{a_{n}[e^{-x}]\\}_{L}.$ (123)

This is analogous to the following relation.

$\displaystyle(e^{x})^{-1}=e^{-x}.$ (124)

As with the left sequence version of Maclaurin series for
$\\{a_{n}[e^{x}]\\}_{R}$, the left sequence version of Maclaurin series for
$\\{a_{n}[e^{-x}]\\}_{L}$ can be obtained. First, the Maclaurin series for the
$e^{-x}$ is expressed as follows.

$\displaystyle e^{-x}$
$\displaystyle=\sum_{n=0}^{\infty}\frac{(-1)^{n}x^{n}}{n!}=\frac{1}{0!}1-\frac{1}{1!}x+\frac{1}{2!}x^{2}-\frac{1}{3!}x^{3}+\cdots$
(125)
$\displaystyle=1-x+\int_{0}^{x}x-\int_{0}^{x}\int_{0}^{x}x+\int_{0}^{x}\int_{0}^{x}\int_{0}^{x}x-\cdots.$
(126)

Then, the left sequence version of Maclaurin series for
$\\{a_{n}[e^{-x}]\\}_{L}$ is obtained as follows.

######  Proposition 9 (Left sequence version of Maclaurin series for left
sequence version of negative exponential function 1).

$\displaystyle\\{a_{n}[e^{-x}]\\}_{L}$
$\displaystyle=\\{a_{n}[1]\\}-\frac{1}{2}\\{a_{n}[x]\\}+\frac{1}{2^{2}}\mathcal{I}_{L}^{0}\\{a_{n}[x]\\}$
$\displaystyle\qquad-\frac{1}{2^{3}}\mathcal{I}_{L}^{0}\mathcal{I}_{L}^{0}\\{a_{n}[x]\\}+\frac{1}{2^{4}}\mathcal{I}_{L}^{0}\mathcal{I}_{L}^{0}\mathcal{I}_{L}^{0}\\{a_{n}[x]\\}-\cdots.$
(127)

By using $\\{a_{n}[x^{n}/n!]\\}_{L}$, we can rewrite Prop. 9 as follows.

######  Proposition 10 (Left sequence version of Maclaurin series for left
sequence version of negative exponential function 2).

$\displaystyle\\{a_{n}[e^{-x}]\\}_{L}=\sum_{n=0}^{\infty}\frac{(-1)^{n}}{2^{n}}\\{a_{n}[x^{n}/n!]\\}_{L}.$
(128)

Also, by using the left integral, we obtain the following proposition of the
right sequence version of Maclaurin series for $\\{a_{n}[e^{-x}]\\}_{L}$.

######  Proposition 11 (Right sequence version of Maclaurin series for left
sequence version of negative exponential function 1).

$\displaystyle\\{a_{n}[e^{-x}]\\}_{L}$
$\displaystyle=\\{a_{n}[1]\\}-\\{a_{n}[x]\\}+\frac{1}{2^{2}}\mathcal{I}_{L}^{0}\\{a_{n}[x]\\}-\mathcal{I}_{L}^{0}\mathcal{I}_{L}^{0}\\{a_{n}[x]\\}$
$\displaystyle\qquad+\mathcal{I}_{L}^{0}\mathcal{I}_{L}^{0}\mathcal{I}_{L}^{0}\\{a_{n}[x]\\}-\cdots.$
(129)

By using $\\{a_{n}[x^{n}/n!]\\}_{R}$, we can rewrite Prop. 11 as follows.

######  Proposition 12 (Right sequence version of Maclaurin series for left
sequence version of negative exponential function 2).

$\displaystyle\\{a_{n}[e^{-x}]\\}_{L}=\sum_{n=0}^{\infty}(-1)^{n}\\{a_{n}[x^{n}/n!]\\}_{R}.$
(130)

We remark that the sums in Props. 11 and 12 should not be interpreted by the
usual sum but the Abel sum [23]. For example, the first term and second term
of the right sequence version of Maclaurin series for
$\\{a_{n}[e^{-x}]\\}_{L}$ are given as follows.

$\displaystyle 1-1+1-1+1-1+\cdots=\frac{1}{2}=a_{1}[x^{n}/n!]_{R},$ (131)
$\displaystyle 1-2+3-4+5-6\cdots=\frac{1}{4}=a_{2}[x^{n}/n!]_{R}.$ (132)

The first one is called Grandi’s series [24], and would be one of the most
famous divergent series. The right sequence version of Maclaurin series for
all the terms of $\\{a_{n}[e^{-x}]\\}_{L}$ are divergent except the zeroth
term if the usual sum is used. However, by using the Abel sum, value of real
numbers can be assigned to those divergent series. Note that the $n$th term of
$\\{a_{n}[e^{-x}]\\}_{L}$ can be represented as follows.

$\displaystyle\left.a_{n}[e^{-x}]_{L}=\frac{1}{(1+x)^{n}}\right|_{x=1}.$ (133)

If $|x|<1$, the r.h.s. of the above equation is given by

$\displaystyle\frac{1}{(1+x)^{n}}$
$\displaystyle=1-nx+\frac{n(n+1)}{2}x^{2}-\frac{n(n+1)(n+2)}{6}x^{3}+\cdots$
$\displaystyle=1+\sum_{k=1}^{\infty}\frac{n(n+1)\cdots(n+k-1)}{k!}(-x)^{k}.$
(134)

If the above sum is interpreted by the Abel sum, the above formula holds even
if $|x|=1$.

######  Remark 4.

Figure 10 (11) shows the diagram representing the right sequence version of
Maclaurin series for $\\{a_{n}[e^{-x}]\\}_{L}$ (left sequence version of
Maclaurin series for $\\{a_{n}[e^{-x}]\\}_{L}$).

Figure 10: Diagrammatic representation of right sequence version of Maclaurin
series for left sequence version of negative exponential function Figure 11:
Diagrammatic representation of left sequence version of Maclaurin series for
left sequence version of negative exponential function

######  Remark 5.

The $\\{a_{n}[e^{-x}]\\}_{R}$ should satisfy the following properties for any
non-negative integer $n$.

$\displaystyle a_{n+1}-a_{n}$ $\displaystyle=-a_{n},$ (135) $\displaystyle
a_{n+1}$ $\displaystyle=0.$ (136)

Then,

$\displaystyle\\{a_{n}[e^{-x}]\\}_{R}=a,0,0,0,\cdots,$ (137)

where $a$ is a real number.

######  Remark 6.

The sequence $\\{a_{n}[e^{-x}]\\}^{\rm nat}$ naturally corresponding to
$e^{-x}$ from the viewpoint of Maclaurin series is given by

$\displaystyle\\{a_{n}[e^{-x}]\\}^{\rm
nat}=\\{a_{n}[1]\\}-\\{a_{n}[x]\\}+\mathcal{I}_{L}^{0}\\{a_{n}[x]\\}-\mathcal{I}_{L}^{0}\mathcal{I}_{L}^{0}\\{a_{n}[x]\\}$
$\displaystyle\qquad+\mathcal{I}_{L}^{0}\mathcal{I}_{L}^{0}\mathcal{I}_{L}^{0}\\{a_{n}[x]\\}-\cdots=\\{(1-1)^{n}\\}=1,0,0,\cdots,$
(138)

where we assumed $0^{0}=1$.

###  5.3 $e^{\alpha x}$

In this subsection, we generalize the result obtained in the previous two
subsections, and define sequence version of $e^{\alpha x}$ for any real number
$\alpha$.

####  5.3.1 $\\{a_{n}[e^{\alpha x}]\\}_{R},\ \alpha\neq-1$

A sequence satisfying the following relation

$\displaystyle\mathcal{D}_{R}\\{a_{n}[e^{\alpha
x}]\\}_{R}=\alpha\\{a_{n}[e^{\alpha x}]\\}_{R}.$ (139)

is given by

######  Definition 21 (Right sequence version of general exponential
function).

$\displaystyle\\{a_{n}[e^{\alpha
x}]\\}_{R}=1,(1+\alpha),(1+\alpha)^{2},\cdots.$ (140)

Then, the exponential function satisfies the law of exponent as follows for
real numbers $a$ and $b$.

$\displaystyle e^{(a+b)x}=e^{ax}e^{bx}.$ (141)

However, the law of exponent for $\\{a_{n}[e^{\alpha x}]\\}_{R}$ itself does
not hold and instead, the following relation holds.

######  Proposition 13 (Law of exponent for right sequence version of general
exponential function).

For real numbers $\alpha$ and $\beta$,

$\displaystyle\\{a_{n}[e^{(\alpha+\beta+\alpha\beta)x}]\\}_{R}=\\{a_{n}[e^{\alpha
x}]\\}_{R}\\{a_{n}[e^{\beta x}]\\}_{R}.$ (142)

Also, the following four propositions of the sequence versions of Maclaurin
series for general exponential function hold.

######  Proposition 14 (Left sequence version of Maclaurin series for right
sequence version of general exponential function 1).

The sequence version of Maclaurin series for $\\{a_{n}[e^{\alpha x}]\\}_{R}$
is given by

$\displaystyle\\{a_{n}[e^{\alpha x}]\\}_{R}$
$\displaystyle=\\{a_{n}[1]\\}+\alpha\\{a_{n}[x]\\}+\alpha^{2}\mathcal{I}_{L}^{0}\\{a_{n}[x]\\}+\alpha^{3}\mathcal{I}_{L}^{0}\mathcal{I}_{L}^{0}\\{a_{n}[x]\\}$
$\displaystyle\qquad+\alpha^{4}\mathcal{I}_{L}^{0}\mathcal{I}_{L}^{0}\mathcal{I}_{L}^{0}\\{a_{n}[x]\\}+\cdots.$
(143)

######  Proposition 15 (Left sequence version of Maclaurin series for right
sequence version of general exponential function 2).

The sequence version of Maclaurin series for $\\{a_{n}[e^{\alpha x}]\\}_{R}$
is given by

$\displaystyle\\{a_{n}[e^{\alpha x}]\\}_{R}$
$\displaystyle=\sum_{n=0}^{\infty}\alpha^{n}\\{a_{n}[x^{n}/n!]\\}_{L}.$ (144)

######  Proposition 16 (Right sequence version of Maclaurin series for right
sequence version of general exponential function 1).

The sequence version of Maclaurin series for $\\{a_{n}[e^{\alpha x}]\\}_{R}$
is given by

$\displaystyle\\{a_{n}[e^{\alpha x}]\\}_{R}$
$\displaystyle=\\{a_{n}[1]\\}+\frac{\alpha}{1+\alpha}\\{a_{n}[x]\\}+\left(\frac{\alpha}{1+\alpha}\right)^{2}\mathcal{I}_{R}^{0}\\{a_{n}[x]\\}+\left(\frac{\alpha}{1+\alpha}\right)^{3}\mathcal{I}_{R}^{0}\mathcal{I}_{R}^{0}\\{a_{n}[x]\\}$
$\displaystyle\qquad+\left(\frac{\alpha}{1+\alpha}\right)^{4}\mathcal{I}_{R}^{0}\mathcal{I}_{R}^{0}\mathcal{I}_{R}^{0}\\{a_{n}[x]\\}+\cdots.$
(145)

######  Proposition 17 (Right sequence version of Maclaurin series for right
sequence version of general exponential function 2).

The sequence version of Maclaurin series for $\\{a_{n}[e^{\alpha x}]\\}_{R}$
is given by

$\displaystyle\\{a_{n}[e^{\alpha x}]\\}_{R}$
$\displaystyle=\sum_{n=0}^{\infty}\left(\frac{\alpha}{1+\alpha}\right)^{n}\\{a_{n}[x^{n}/n!]\\}_{R}.$
(146)

The right sequence version of Maclaurin series should be interpreted through
Borel summation [23]. The following formula usually holds if $|\alpha|<1$.

$\displaystyle
1+\frac{\alpha}{1+\alpha}+\left(\frac{\alpha}{1+\alpha}\right)^{2}+\cdots=1+\alpha.$
(147)

However, if the r.h.s. is interpreted by the Borel summation, for all
$\alpha\neq-1$, the divergent series appeared in the right sequence version of
Maclaurin series can have value of real numbers. For example, the sequence
version of Maclaurin series for the $a_{1}[e^{-\frac{2}{3}x}]_{R}$ is given by

$\displaystyle 1-2+4-8+16-32+\cdots=\frac{1}{3}=a_{1}[e^{-\frac{2}{3}x}]_{R}.$
(148)

####  5.3.2 $\\{a_{n}[e^{\alpha x}]\\}_{L},\ \alpha\neq 1$

A sequence satisfying the following relation

$\displaystyle\mathcal{D}_{L}\\{a_{n}[e^{\alpha
x}]\\}_{L}=\alpha\\{a_{n}[e^{\alpha x}]\\}_{L},$ (149)

is given by

######  Definition 22 (Left sequence version of general exponential function).

$\displaystyle\\{a_{n}[e^{\alpha
x}]\\}_{L}=1,(1-\alpha)^{-1},(1-\alpha)^{-2},\cdots.$ (150)

Then, the sequence version of the law of exponent of exponential function (Eq.
(141)) is given as follows.

######  Proposition 18 (Law of exponent for left sequence version of general
exponential function).

For real numbers $\alpha$ and $\beta$,

$\displaystyle\\{a_{n}[e^{(\alpha+\beta-\alpha\beta)x}]\\}_{L}=\\{a_{n}[e^{\alpha
x}]\\}_{L}\\{a_{n}[e^{\beta x}]\\}_{L}.$ (151)

Also, we have a generalization of the inverse relation (Eq. 123) as follows.

######  Proposition 19 (Inverse relation for sequence versions of general
exponential function).

For a real number $\alpha$ that is not equals to $-1$,

$\displaystyle\\{a_{n}[e^{\alpha x}]\\}_{R}^{-1}=\\{a_{n}[e^{-\alpha
x}]\\}_{L}.$ (152)

Furthermore, the following four propositions of the sequence versions of
Maclaurin series for general exponential function hold.

######  Proposition 20 (Left sequence version of Maclaurin series for left
sequence version of general exponential function 1).

The sequence version of Maclaurin series for $\\{a_{n}[e^{\alpha x}]\\}_{L}$
is given by

$\displaystyle\\{a_{n}[e^{\alpha x}]\\}_{L}$
$\displaystyle=\\{a_{n}[1]\\}+\frac{\alpha}{1-\alpha}\\{a_{n}[x]\\}+\left(\frac{\alpha}{1-\alpha}\right)^{2}\mathcal{I}_{L}^{0}\\{a_{n}[x]\\}$
$\displaystyle\qquad+\left(\frac{\alpha}{1-\alpha}\right)^{3}\mathcal{I}_{L}^{0}\mathcal{I}_{L}^{0}\\{a_{n}[x]\\}+\left(\frac{\alpha+1}{\alpha}\right)^{4}\mathcal{I}_{L}^{0}\mathcal{I}_{L}^{0}\mathcal{I}_{L}^{0}\\{a_{n}[x]\\}+\cdots.$
(153)

######  Proposition 21 (Left sequence version of Maclaurin series for left
sequence version of general exponential function 2).

The sequence version of Maclaurin series for $\\{a_{n}[e^{\alpha x}]\\}_{R}$
is given by

$\displaystyle\\{a_{n}[e^{\alpha x}]\\}_{L}$
$\displaystyle=\sum_{n=0}^{\infty}\left(\frac{\alpha}{1-\alpha}\right)^{n}\\{a_{n}[x^{n}/n!]\\}_{L}.$
(154)

######  Proposition 22 (Right sequence version of Maclaurin series for left
sequence version of general exponential function 1).

The sequence version of Maclaurin series for $\\{a_{n}[e^{\alpha x}]\\}_{R}$
is given by

$\displaystyle\\{a_{n}[e^{\alpha x}]\\}_{L}$
$\displaystyle=\\{a_{n}[1]\\}+\alpha\\{a_{n}[x]\\}+\alpha^{2}\mathcal{I}_{R}^{0}\\{a_{n}[x]\\}+\alpha^{3}\mathcal{I}_{R}^{0}\mathcal{I}_{R}^{0}\\{a_{n}[x]\\}$
$\displaystyle\qquad+\alpha^{4}\mathcal{I}_{R}^{0}\mathcal{I}_{R}^{0}\mathcal{I}_{R}^{0}\\{a_{n}[x]\\}+\cdots.$
(155)

######  Proposition 23 (Right sequence version of Maclaurin series for left
sequence version of general exponential function 2).

The sequence version of Maclaurin series for $\\{a_{n}[e^{\alpha x}]\\}_{R}$
is given by

$\displaystyle\\{a_{n}[e^{\alpha x}]\\}_{L}$
$\displaystyle=\sum_{n=0}^{\infty}\alpha^{n}\\{a_{n}[x^{n}/n!]\\}_{R}.$ (156)

##  6 Hyperbolic function

In this subsection, we define the sequence version of hyperbolic function in
two ways. Because the hyperbolic cosine function and hyperbolic sine function
are defined as follows.

$\displaystyle\cosh
x=\frac{e^{x}+e^{-x}}{2}=\sum_{n=0}^{\infty}\frac{x^{2n}}{(2n)!},$ (157)
$\displaystyle\sinh
x=\frac{e^{x}-e^{-x}}{2}=\sum_{n=0}^{\infty}\frac{x^{2n+1}}{(2n+1)!},$ (158)

and we have already defined the sequence version of the $e^{x}$ in Eq. (94)
and $e^{-x}$ in Eq. (122), then, we can obtain sequence version of the
hyperbolic cosine function and hyperbolic sine function as follows.

######  Definition 23 (Sequence version of hyperbolic function).

$\displaystyle\\{a_{n}[\cosh
x]\\}=\frac{1}{2}[\\{a_{n}[[e^{x}]]\\}_{R}+\\{a_{n}[e^{-x}]\\}_{L}]=\frac{1}{2}\\{2^{n}+2^{-n}\\}=1,\frac{5}{4},\frac{17}{8},\frac{65}{16},\cdots,$
(159) $\displaystyle\\{a_{n}[\sinh
x]\\}=\frac{1}{2}[\\{a_{n}[[e^{x}]]\\}_{R}-\\{a_{n}[e^{-x}]\\}_{L}]=\frac{1}{2}\\{2^{n}-2^{-n}\\}=0,\frac{3}{4},\frac{15}{8},\frac{63}{16},\cdots.$
(160)

Figure 12(a) shows difference between the usual hyperbolic cosine function
($\cosh x$) and the sequence version of the hyperbolic cosine function
($\\{a_{n}[\cosh x]\\}$) and Fig. 12(b) shows difference between the usual
hyperbolic sine function ($\sinh x$) and the sequence version of the
hyperbolic sine function ($\\{a_{n}[\sinh x]\\}$).

Figure 12: Difference between (a) $\cosh x$ and $\\{a_{n}[\cosh x]\\}$, (b)
$\sinh x$ and $\\{a_{n}[\sinh x]\\}$, (c) $\\{a_{n}[\cosh x]\\}$
($\\{a_{n}[\sinh x]\\}^{\rm nat}$) and $\\{a_{n}[\cosh x]\\}^{\rm nat}$
($\\{a_{n}[\sinh x]\\}$).

The hyperbolic cosine function and hyperbolic sine function have the following
properties.

$\displaystyle\frac{d}{dx}\cosh x$ $\displaystyle=\sinh x,$ (161)
$\displaystyle\frac{d}{dx}\sinh x$ $\displaystyle=\cosh x.$ (162)

Then, analogously, we find that $\\{a_{n}[\cosh x]\\}$ and $\\{a_{n}[\sinh
x]\\}$ have the following properties.

$\displaystyle\mathcal{D}_{R}\\{a_{n}[\cosh x]\\}$
$\displaystyle=\\{(2^{2n+1}-1)/2^{n+2}\\}=\frac{1}{4},\frac{7}{8},\frac{31}{16},\cdots=\\{a_{n}[\sinh
x]\\}+\\{1/(2^{n+2})\\},$ (163) $\displaystyle\mathcal{D}_{R}\\{a_{n}[\sinh
x]\\}$
$\displaystyle=\\{(2^{2n+1}+1)/2^{n+2}\\}=\frac{3}{4},\frac{9}{8},\frac{33}{16},\cdots=\\{a_{n}[\cosh
x]\\}-\\{1/(2^{n+2})\\}.$ (164)

Also, the following relation holds for the hyperbolic functions.

$\displaystyle\cosh^{2}x-\sinh^{2}x=1.$ (165)

Analogously, the following relation holds for $\\{a_{n}[\cosh x]\\}$ and
$\\{a_{n}[\sinh x]\\}$.

$\displaystyle\\{a_{n}[\cosh x]\\}^{2}-\\{a_{n}[\sinh x]\\}^{2}=\\{1\\}.$
(166)

Furthermore, there exists an addition theorem for the hyperbolic functions as
follows.

$\displaystyle\cosh^{2}x+\sinh^{2}x=\cosh 2x,$ (167) $\displaystyle 2\cosh
x\sinh x=\sinh 2x.$ (168)

Analogously, the following relation holds for $\\{a_{n}[\cosh x]\\}$ and
$\\{a_{n}[\sinh x]\\}$.

######  Proposition 24 (Sequence version of addition theorem for hyperbolic
function).

$\displaystyle\\{a_{n}[\cosh x]\\}^{2}+\\{a_{n}[\sinh
x]\\}^{2}=\\{a_{2n}[\cosh x]\\},$ (169) $\displaystyle 2\\{a_{n}[\cosh
x]\\}\\{a_{n}[\sinh x]\\}=\\{a_{2n}[\sinh x]\\}.$ (170)

A sequence version of hyperbolic function can also be defined as follows by
using Eq. (138).

######  Definition 24 (Sequence version of hyperbolic function (natural)).

$\displaystyle\\{a_{n}[\cosh x]\\}^{\rm
nat}=\frac{1}{2}[\\{a_{n}[[e^{x}]]\\}+\\{a_{n}[e^{-x}]\\}^{\rm
nat}]=1,1,2,4,8,\cdots,$ (171) $\displaystyle\\{a_{n}[\sinh x]\\}^{\rm
nat}=\frac{1}{2}[\\{a_{n}[[e^{x}]]\\}-\\{a_{n}[e^{-x}]\\}^{\rm
nat}]=0,1,2,4,8,\cdots.$ (172)

As an analogy of Eqs. (161) and (162), $\\{a_{n}[\cosh x]\\}^{\rm nat}$ and
$\\{a_{n}[\sinh x]\\}^{\rm nat}$ have the following properties.

$\displaystyle\mathcal{D}_{R}\\{a_{n}[\cosh x]\\}^{\rm nat}$
$\displaystyle=\\{a_{n}[\sinh x]\\}^{\rm nat},$ (173)
$\displaystyle\mathcal{D}_{R}\\{a_{n}[\sinh x]\\}^{\rm nat}$
$\displaystyle=\\{a_{n}[\cosh x]\\}^{\rm nat}.$ (174)

Also, $\\{a_{n}[\cos x]\\}^{\rm nat}$ and $\\{a_{n}[\sin x]\\}^{\rm nat}$ have
the following property analogous to Eq. (165).

$\displaystyle(\\{a_{n}[\cosh x]\\}^{\rm nat})^{2}-(\\{a_{n}[\sinh x]\\}^{\rm
nat})^{2}=\\{a_{n}[e^{-x}]\\}^{\rm nat}.$ (175)

Furthermore, the $\\{a_{n}[\cosh x]\\}^{\rm nat}$ and $\\{a_{n}[\sinh
x]\\}^{\rm nat}$ satisfy the following analogy of the addition theorem (Eq.
(167) and (168)).

######  Proposition 25 (Sequence version of addition theorem for hyperbolic
function (natural)).

$\displaystyle(\\{a_{n}[\cosh x]\\}^{\rm nat})^{2}+(\\{a_{n}[\sinh x]\\}^{\rm
nat})^{2}=\\{a_{2n}[\cosh x]\\}^{\rm nat},$ (176) $\displaystyle
2\\{a_{n}[\cosh x]\\}^{\rm nat}\\{a_{n}[\sinh x]\\}^{\rm nat}=\\{a_{2n}[\sinh
x]\\}^{\rm nat}.$ (177)

Figure 12(c) shows difference between $\\{a_{n}[\cosh x]\\}$ ($\\{a_{n}[\sinh
x]\\}^{\rm nat}$) and $\\{a_{n}[\cosh x]\\}^{\rm nat}$ ($\\{a_{n}[\sinh
x]\\}$).

##  7 Trigonometric function

First, we define sequence version of trigonometric function in two ways. By
extending domain of Eq. (140) from real numbers to complex numbers and
focusing on the following function relation

$\displaystyle e^{ix}=\cos x+i\sin x,$ (178)

we obtain the following sequence version of the cosine function and sin
function.

######  Definition 25 (Right sequence version of trigonometric function).

$\displaystyle\\{a_{n}[\cos x]\\}_{R}$ $\displaystyle=\\{{\rm
Re}(1+i)^{n}\\}=1,1,0,-2,-4,-4,0,8,16,16,\cdots,$ (179)
$\displaystyle\\{a_{n}[\sin x]\\}_{R}$ $\displaystyle=\\{{\rm
Im}(1+i)^{n}\\}=0,1,2,2,0,-4,-8,-8,0,16,\cdots.$ (180)

Figure 13(a) shows a graph of $(\\{a_{n}[\cos x]\\}_{R},\\{a_{n}[\sin
x]\\}_{R})$ that represents the divergent spiral. This contrasts with the
circle represented by $(\cos x,\sin x)$.

Figure 13: Graph of (a) $(\\{a_{n}[\cos x]\\}_{R},\\{a_{n}[\sin x]\\}_{R})$
corresponding to the divergent spiral and (b) $(\\{a_{n}[\cos
x]\\}_{L},\\{a_{n}[\sin x]\\}_{L})$ corresponding to the convergent spiral.

The cosine function and sine function have the following properties for
differential.

$\displaystyle\frac{d}{dx}\cos x$ $\displaystyle=-\sin x,$ (181)
$\displaystyle\frac{d}{dx}\sin x$ $\displaystyle=\cos x.$ (182)

Then, analogously, we find that $\\{a_{n}[{\rm cos}\,x]\\}_{R}$ and
$\\{a_{n}[\sin x]\\}_{R}$ have the following properties.

$\displaystyle\mathcal{D}_{R}\\{a_{n}[\cos x]\\}_{R}$
$\displaystyle=0,-1,-2,-2,0,4,8,8,0,-16,\cdots=-\\{a_{n}[\sin x]\\}_{R},$
(183) $\displaystyle\mathcal{D}_{R}\\{a_{n}[\sin x]\\}_{R}$
$\displaystyle=1,1,0,-2,-4,-4,0,8,16,16,\cdots=\\{a_{n}[\cos x]\\}_{R}.$ (184)

Also, the following relation holds for the trigonometric functions.

$\displaystyle\cos^{2}x+\sin^{2}x=1.$ (185)

Analogously, the following formula for $\\{a_{n}[\cos x]\\}_{R}$ and
$\\{a_{n}[\sin x]\\}_{R}$ holds.

$\displaystyle\\{a_{n}[\cos x]\\}_{R}^{2}+\\{a_{n}[\sin x]\\}_{R}^{2}=2^{n}.$
(186)

In addition, the tangent function is given by

$\displaystyle\tan x=\frac{\sin x}{\cos x}.$ (187)

Then, the sequence version of the tangent function determined by
$\\{a_{n}[\cos x]\\}_{R}$ and $\\{a_{n}[\sin x]\\}_{R}$ is given by

$\displaystyle\\{a_{n}[\tan x]\\}_{R}=\frac{\\{a_{n}[\sin
x]\\}_{R}}{\\{a_{n}[\cos x]\\}_{R}}=0,1,\infty,-1,0,1,-\infty,-1,0,1,\cdots.$
(188)

where we wrote $n/0=\infty$ and $-n/0=-\infty$ for a natural number $n$.

Furthermore, the trigonometric functions satisfy an addition theorem as
follows.

$\displaystyle\cos 2x$ $\displaystyle=\cos^{2}x-\sin^{2}x,$ (189)
$\displaystyle\sin 2x$ $\displaystyle=2\sin x\cos x.$ (190)

Then, we find the direct analogy of the addition theorem for trigonometric
function as follows.

######  Proposition 26 (Addition theorem for right sequence version of
trigonometric function).

$\displaystyle\\{a_{n}[\cos x]\\}_{R}^{2}-\\{a_{n}[\sin x]\\}_{R}^{2}$
$\displaystyle=1,0,-4,0,16,0,\cdots=\\{a_{2n}[\cos x]\\}_{R},$ (191)
$\displaystyle 2\\{a_{n}[\cos x]\\}_{R}\\{a_{n}[\sin x]\\}_{R}$
$\displaystyle=0,2,0,-8,0,32,\cdots=\\{a_{2n}[\sin x]\\}_{R}.$ (192)

By using Eq. (150), one has another definition of sequence version of cosine
function and sine function as follows.

######  Definition 26 (Left sequence version of trigonometric function).

$\displaystyle\\{a_{n}[\cos x]\\}_{L}$ $\displaystyle=\\{{\rm
Re}(1-i)^{-n}\\}=1,\frac{1}{2},0,-\frac{1}{4},-\frac{1}{4},-\frac{1}{8},0,\frac{1}{16}\frac{1}{16},\frac{1}{32},\cdots,$
(193) $\displaystyle\\{a_{n}[\sin x]\\}_{L}$ $\displaystyle=\\{{\rm
Im}(1-i)^{-n}\\}=0,\frac{1}{2},\frac{1}{2},\frac{1}{4},0,-\frac{1}{8},-\frac{1}{8},-\frac{1}{16},0,\frac{1}{32},\cdots.$
(194)

Figure 13(b) shows a graph of $(\\{a_{n}[\cos x]\\}_{L},\\{a_{n}[\sin
x]\\}_{L})$ that represents the convergent spiral. This contrasts with the
circle represented by $(\cos x,\sin x)$, and also the divergent spiral
represented by $(\\{a_{n}[\cos x]\\}_{R},\\{a_{n}[\sin x]\\}_{R})$.

As an analogy of Eqs. (181) and (182), $\\{a_{n}[\cos x]\\}_{L}$ and
$\\{a_{n}[\sin x]\\}_{L}$ have the following properties.

$\displaystyle\mathcal{D}_{L}\\{a_{n}[\cos x]\\}_{L}$
$\displaystyle=0,-\frac{1}{2},-\frac{1}{2},-\frac{1}{4},0,\frac{1}{8},\frac{1}{8},\frac{1}{16},0,-\frac{1}{32},\cdots=-\\{a_{n}[\sin
x]\\}_{L},$ (195) $\displaystyle\mathcal{D}_{L}\\{a_{n}[\sin x]\\}_{L}$
$\displaystyle=1,\frac{1}{2},0,-\frac{1}{4},-\frac{1}{4},-\frac{1}{8},0,\frac{1}{16}\frac{1}{16},\frac{1}{32},\cdots=\\{a_{n}[\cos
x]\\}_{L},$ (196)

where we assumed $a_{-1}[\cos x]_{L}=1$ and $a_{-1}[\sin x]_{L}=-1$.

Also, $\\{a_{n}[\cos x]\\}_{L}$ and $\\{a_{n}[\sin x]\\}_{L}$ have the
following property that is analogous to Eq. (185).

$\displaystyle\\{a_{n}[\cos x]\\}_{L}^{2}+\\{a_{n}[\sin x]\\}_{L}^{2}=2^{-n}.$
(197)

In addition, the left sequence version of the tangent function is given by

$\displaystyle\\{a_{n}[\tan x]\\}_{L}=\frac{\\{a_{n}[\sin
x]\\}_{L}}{\\{a_{n}[\cos
x]\\}_{L}}=0,1,\infty,-1,0,1,-\infty,-1,0,1,\cdots=\\{a_{n}[\tan x]\\}_{R}.$
(198)

Furthermore, $\\{a_{n}[\cos x]\\}_{L}$ and $\\{a_{n}[\sinh x]\\}_{L}$ satisfy
the following formula that is analogous to the addition theorem (Eq. (189) and
(190)).

######  Proposition 27 (Addition theorem for left sequence version of
trigonometric function).

$\displaystyle\\{a_{n}[\cos x]\\}_{L}^{2}-\\{a_{n}[\sin x]\\}_{L}^{2}$
$\displaystyle=1,0,-\frac{1}{4},0,\frac{1}{16},0,\cdots=\\{a_{2n}[\cos
x]\\}_{L},$ (199) $\displaystyle 2\\{a_{n}[\cos x]\\}_{L}\\{a_{n}[\sin
x]\\}_{L}$
$\displaystyle=0,\frac{1}{2},0,-\frac{1}{8},0,\frac{1}{32},\cdots=\\{a_{2n}[\sin
x]\\}_{L}.$ (200)

The following equality is called Euler’s identity and connects the Napier’s
constant ($e$), the additive identity ($0$), the multiplicative identity
($1$), the imaginary unit ($i$), the circle ratio ($\pi$) , and would be one
of the most famous and beautiful equality in mathematics.

$\displaystyle e^{i\pi}+1=0.$ (201)

Note that this equation can be rewritten in two ways as follows.

$\displaystyle e^{i\pi}+1^{2}=0,$ (202) $\displaystyle e^{i\pi}+1^{-2}=0.$
(203)

Then, we construct sequence version of the Euler’s identity by using Eq. (140)
and Eq. (150) as follows.

######  Theorem 2 (Sequence version of Euler’s identity).

$\displaystyle a_{4}[e^{ix}]_{R}+2^{2}=0,$ (204) $\displaystyle
a_{4}[e^{ix}]_{L}+2^{-2}=0.$ (205)

Figure 14(a) shows schematic representation of Eqs. (204) and (205).

Figure 14: (a) Sequence version of Euler’s identity
($a_{4}[e^{ix}]_{R}+2^{2}=0$ and $a_{4}[e^{ix}]_{L}+2^{-2}=0$), (b)
$a_{4}[e^{ix}]_{L}a_{4}[e^{ix}]_{R}=1.$

Equation (204) is obtained by replacing $e^{i\pi}$ by $a_{4}[e^{ix}]_{R}$ and
$1^{2}$ by $2^{2}$ in Eq. (202). Also, Eq. (205) is obtained by replacing
$e^{i\pi}$ by $a_{4}[e^{ix}]_{L}$ and $1^{-2}$ by $2^{-2}$ in Eq. (203). In
addition, as an analogy of the relation (see Fig. 3(c))

$\displaystyle e^{i\pi}e^{i\pi}=1,$ (206)

the following formula holds.

$\displaystyle a_{4}[e^{ix}]_{L}a_{4}[e^{ix}]_{R}=1.$ (207)

Figure 14(b) shows schematic representation of Eq. (207).

Also, by combining $\\{a_{n}[\cos x]\\}_{R}$ ($\\{a_{n}[\sin x]\\}_{L}$) and
$\\{a_{n}[\cos x]\\}_{L}$ ($\\{a_{n}[\sin x]\\}_{L}$) generally, one has the
following sequence version of $\cos x$ and $\sin x$ whose periodicity is 8.

$\displaystyle\\{a_{n}[\cos x]\\}^{\rm per}$ $\displaystyle=\\{{\rm sgn}\
(a_{n}[\cos x]_{R})\\}\times\sqrt{\\{a_{n}[\cos x]\\}_{R}\times\\{a_{n}[\cos
x]\\}_{L}}$ (208)
$\displaystyle=1,\frac{1}{\sqrt{2}},0,-\frac{1}{\sqrt{2}},-1,-\frac{1}{\sqrt{2}},0,\frac{1}{\sqrt{2}},\cdots,$
(209) $\displaystyle\\{a_{n}[\sin x]\\}^{\rm per}$ $\displaystyle=\\{{\rm
sgn}\ (a_{n}[\sin x]_{R})\\}\times\sqrt{\\{a_{n}[\sin
x]\\}_{R}\times\\{a_{n}[\sin x]\\}_{L}}$ (210)
$\displaystyle=0,\frac{1}{\sqrt{2}},1,\frac{1}{\sqrt{2}},0,-\frac{1}{\sqrt{2}},-1,-\frac{1}{\sqrt{2}},\cdots.$
(211)

This periodicity is analogous to $2\pi$-periodicity of $\cos x$ and $\sin x$.
In Fig. 15, we show representation of $(\cos x,\sin x)$ by the unit circle and
of $(a_{n}[\cos x]^{\rm per},a_{n}[\sin x]^{\rm per})$ by the eight points on
the unit circle. While the $2\pi$ is the periodicity of $\cos x$ and $\sin x$
and the circumference of a unit circle, and the 8 is the periodicity of
$\\{a_{n}[\cos x]\\}^{\rm per}$ and $\\{a_{n}[\sin x]\\}^{\rm per}$ and the
length of the circumference of the unit circle when the unit is given by the
distance between two adjacent points on the unit circle ($\pi/4$).

Figure 15: (a) Representation of $(\cos x,\sin x)$ by the unit circle. (b)
Representation of $(a_{n}[\cos x]^{\rm per},a_{n}[\sin x]^{\rm per})$ by the
eight points on the unit circle.

##  8 Fibonacci sequences

In this section, we apply the sequence version of Maclaurin series to the
Fibonacci sequence and its generalizations.

###  8.1 Fibonacci sequence

Fibonacci sequence is defined by

$\displaystyle\\{F_{n}\\}=0,1,1,2,3,5,8,\cdots,$ (212)

and satisfies the relation

$\displaystyle\lim_{n\to\infty}\frac{F_{n}}{F_{n-1}}=\phi,$ (213)

where $\phi$ is the golden ratio and also the relation

$\displaystyle F_{n+2}=F_{n+1}+F_{n}.$ (214)

This relation can be rewritten by using the right differential
$\mathcal{D}_{R}$ defined in Eq. (52) as follows.

$\displaystyle\mathcal{D}_{R}\\{F_{n+1}\\}=\\{F_{n}\\}.$ (215)

Also, by using the shift operator $\mathcal{S}_{k}$ defined in Eq. (42), one
has

$\displaystyle\mathcal{D}_{R}\mathcal{S}_{1}\\{F_{n}\\}=\\{F_{n}\\}.$ (216)

This is similar to Eq. (91) that the right sequence version of exponential
function satisfies. Then, we obtain the following theorem by using the
insertion operator $I_{a}$ defined in Eq. (50) that is similar to Eq. (4) (see
Fig. 16 and compare Fig. 17 with Fig. 8).

######  Theorem 3 (Sequence version of Maclaurin series for Fibonacci
sequence).

$\displaystyle\\{F_{n}\\}=I_{0}\\{a_{n}[1]\\}+I_{0}^{2}\\{a_{n}[x]\\}+I_{0}^{3}\\{a_{n}[x^{2}/2]\\}+\cdots=\sum_{n=0}^{\infty}I_{0}^{n+1}\\{a_{n}[x^{n}/n!]\\}.$
(217)

Figure 16: Digrammatic representation of sequence version of Maclaurin series
for Fibonacci sequence corresponding to Pascal’s triangle. Figure 17:
Diagrammatic representation of sequence version of Maclaurin series for
Fibonacci sequence by using blocks.

Also, the negafibonacci sequence [25] is defined by

$\displaystyle\\{F_{n}^{-}\\}=0,1,-1,2,-3,5,-8,\cdots=\\{F_{n}\\}\times\\{(-1)^{n+1}\\},$
(218)

and satisfies the following relation.

$\displaystyle\mathcal{D}_{R}\\{F_{n}^{-}\\}=-\mathcal{S}_{2}\\{F_{n}^{-}\\}.$
(219)

Although this is similar to $e^{-x}$, however the relation
$\\{F_{n}\\}\times\\{F_{n}^{-}\\}=\\{a_{n}[1]\\}$ does not hold. Instead the
following relation holds that is equivalent to Cassini’s Fibonacci identity
[26].

$\displaystyle\\{F_{n}\\}\times\mathcal{S}_{2}\\{F_{n}^{-}\\}+\mathcal{S}_{1}\\{F_{n}\\}\times\mathcal{S}_{1}\\{F_{n}^{-}\\}=\\{a_{n}[1]\\}.$
(220)

For example, $F_{1}F_{3}^{-}+F_{2}^{-}F_{2}=1\times 2+1\times(-1)=1$.

###  8.2 $(P,\,Q)$-Fibonacci sequence

The $(P,\,Q)$-Fibonacci sequence is given by

$\displaystyle\\{F_{n}(P,Q)\\}=0,1,P,P^{2}+Q,P^{3}+2PQ,P^{4}+3P^{2}Q+Q^{2},\cdots,$
(221)

and is a generalization of Fibonacci sequence and satisfies the following
relation [27, 28].

$\displaystyle\\{F_{n+2}(P,Q)\\}=P\\{F_{n+1}(P,Q)\\}+Q\\{F_{n}(P,Q)\\}.$ (222)

We find the $(P,\,Q)$-Fibonacci sequence has the following sequence version of
Maclaurin series (Fig. 18).

######  Proposition 28 (Sequence version of Maclaurin series for
$(P,Q)$-Fibonacci sequence).

$\displaystyle\\{F_{n}(P,Q)\\}$
$\displaystyle=I_{0}[\\{a_{n}[1]\\}\times\\{P^{n}\\}]+QI_{0}^{2}[\\{a_{n}[x]\\}\times\\{P^{n}\\}]+\cdots,$
(223)
$\displaystyle=\sum_{k=0}^{\infty}I_{0}^{k+1}Q^{k}[\\{a_{n}[x^{k}/k!]\\}\times\\{P^{n}\\}].$
(224)

Figure 18: Digrammatic representation of sequence version of Maclaurin series
for $(P,\ Q)$-Fibonacci sequence.

####  8.2.1 Example 1: Pell sequence ($P=2$, $Q=1$)

The $(2,1)$-Fibonacci sequence is called Pell sequence [29, 30, 31, 32] and
satisfies the following relation.

$\displaystyle\lim_{n\to\infty}\frac{F_{n}(2,1)}{F_{n-1}(2,1)}=1+\sqrt{2},$
(225)

where $1+\sqrt{2}$ is the silver ratio. The sequence version of Maclaurin
series for Pell Sequence is shown in Fig. 19.

Figure 19: Digrammatic representation of sequence version of Maclaurin series
for $(2,\ 1)$-Fibonacci sequence (Pell sequence).

####  8.2.2 Example 2: Jacobsthal sequence ($P=1$, $Q=2$)

The $(1,2)$-Fibonacci sequence is called Jacobsthal sequence [33, 34, 35] and
the sequence version of Maclaurin series for Jacobsthal Sequence is shown in
Fig. 20.

Figure 20: Digrammatic representation of sequence version of Maclaurin series
for $(1,\ 2)$-Fibonacci sequence (Jacobsthal sequence).

###  8.3 $k$-bonacci sequence

In this subsection, we propose the sequence version of $k$-bonacci sequence
[36] satisfying the following relation.

$\displaystyle\\{F_{n+k}(k)\\}=\sum_{l=0}^{k-1}\\{F_{n+l}(k)\\}.$ (226)

Then, we construct the sequence version of Maclaurin series for
$\\{F_{n+k}(k)\\}$. First, we define deformed integral by combining the
integral $\mathcal{I}_{0}$ and the insertion $I_{0}$.

######  Definition 27 (Deformed integral).

For an integer $k\geq 2$,

$\displaystyle\tilde{\mathcal{I}}_{k}=\sum_{l=0}^{k-2}I_{0}^{l}\mathcal{I}_{L}^{0}.$
(227)

Then, we have the following proposition.

######  Proposition 29 (Sequence version of Maclaurin series for $k$-bonacci
sequence).

For an integer $k$ such that $k\geq 2$,

$\displaystyle\\{F_{n}(k)\\}$
$\displaystyle=I_{0}^{k-1}\\{a_{n}[1]\\}+\tilde{\mathcal{I}}_{k}I_{0}^{k}\\{a_{n}[1]\\}+\tilde{\mathcal{I}}_{k}^{2}I_{0}^{k+1}\\{a_{n}[1]\\}+\cdots$
(228)
$\displaystyle=\sum_{l=0}^{\infty}(\tilde{\mathcal{I}}_{k})^{l}I_{0}^{k-1+l}\\{a_{n}[1]\\}.$
(229)

This is a generalization of the left sequence version of Maclaurin series for
right sequence version of exponential function (Prop. 2). Also, we define the
limit of sequence as follows.

######  Definition 28 (Limit of sequence).

For an integer $k$ and a sequence $\\{a_{n}\\}$ depending on $k$,

$\displaystyle\lim_{k\to\infty}\\{a_{n}(k)\\}=\lim_{k\to\infty}a_{0}(k),\lim_{k\to\infty}a_{1}(k),\cdots.$
(230)

Then, we obtain the following relation that states
$\mathcal{S}_{k}\\{F_{n}(k)\\}$ approaches to $\\{a_{n}[e^{x}]\\}_{R}$ in the
limit of $k\to\infty$.

######  Proposition 30 (Limit of $k$-bonacci sequence and right sequence
version of exponential function).

$\displaystyle\lim_{k\to\infty}\mathcal{S}_{k}\\{F_{n}(k)\\}=\\{a_{n}[e^{x}]\\}_{R}.$
(231)

We show sequence version of Maclaurin series for tribonacci sequence
($\\{F_{n}(3)\\}$) in Fig. 21. For example, the following relation holds.

$\displaystyle\tilde{\mathcal{I}}_{2}I_{0}^{3}\\{a_{n}[1]\\}$
$\displaystyle=\mathcal{I}_{L}^{0}[I_{0}^{3}\\{a_{n}[1]\\}+I_{0}^{4}\\{a_{n}[1]\\}]=\mathcal{I}_{L}^{0}(0,0,0,1,2,2,2,\cdots)$
$\displaystyle=0,0,0,0,1,3,5,7,\cdots.$ (232)

Figure 21: Digrammatic representation of sequence version of Maclaurin series
for tribonacci sequence ($\\{F_{n}(3)\\}$).

##  9 Factorial and Bell number

In this section, we construct sequence dual of factorials and Bell numbers and
propose sequence dual of the modular property of factorial concerning prime
number and of Bell numbers concerning prime number.

###  9.1 Factorial

Factorial is defined for a non-negative number $n$ as follows.

$\displaystyle n!=n\times(n-1)\times\cdots\times 2\times 1,$ (233)

and $0!=1$. Because the factorial has the following integral representation

$\displaystyle n!=\int_{0}^{\infty}x^{n-1}e^{-x}dx,$ (234)

we construct sequence dual of factorial as follows.

$\displaystyle\\{a_{n}\langle n!\rangle\\}$
$\displaystyle=\sum_{k=0}^{\infty}k^{n}\times 2^{-k},$ (235)
$\displaystyle=2,2,6,26,150,1082,9366,94586,1091670,14174522,\cdots.$ (236)

We find that $\\{a_{n}\langle n!\rangle\\}$ deeply relates to the factorial
$n!$. First, we focus on the following transform for a sequence $\\{a_{n}\\}$
called Stirling transform [37].

$\displaystyle\\{b_{n}\\}=\left\\{\sum_{k=0}^{n}S(n,k)a_{k}\right\\},$ (237)

where $S(n,k)$ is the Stirling number of the second kind that counts how many
ways to partition a set of $n$ elements into $k$ non-empty subsets. For
example, $S(3,2)=3$ because $\\{a,b,c\\}$ has the following subgroups that has
two elements: $\\{a,b\\}$, $\\{b,c\\}$, $\\{c,a\\}$.

Then, the following proposition holds.

######  Proposition 31 (Sequence dual of factorial and Stirling transform).

$\displaystyle\mathcal{S}_{1}\\{a_{n}\langle
n!\rangle\\}=2\left\\{\sum_{k=0}^{n}S(n,k)k!\right\\}.$ (238)

This proposition shows the sequence dual of factorial is equivalent to the
Stirling transform of the second kind for the factorial up to the shift
operation and scalar multiplication.

Next, we focus on the following property of the factorial concerning the prime
numbers called Wilson’s theorem. For a natural number $n$,

$\displaystyle n!\equiv-1\ ({\rm mod}\ n+1),$ (239)

if and only if $n+1$ is a prime. For example, $4\times 3\times 2\times
1=24=5\times 5-1$, $5\times 4\times 3\times 2\times 1=120=6\times 20-0$. Then,
we obtain the sequence dual of Wilson’s theorem as follows.

######  Theorem 4 (Sequence dual of Wilson’s theorem).

For a natural number $n$,

$\displaystyle\\{a_{n}\langle n!\rangle\\}\equiv 0\ ({\rm mod}\ {n+1}),$ (240)

if $n+1$ is a prime number.

For example, $\\{a_{6}\langle n!\rangle\\}=9366=1338\times 7$ and
$\\{a_{8}\langle n!\rangle\\}=1091670=121297\times 9-3$. To prove the above
theorem, it is enough to see the following formula holds for a prime number
$n$ and a natural number $k$ that is less than or equal to $n-1$,

$\displaystyle k!\times S_{n-1,k}\equiv-(-1)^{k}\ {\rm mod}\ n,$ (241)

because the above formula leads to $\sum_{k=1}^{n-1}k!\times S_{n-1,k}\equiv
0\ {\rm mod}\ n$. By using the following formula,

$\displaystyle k!\times
S_{n-1,k}=\sum_{i=0}^{k}(-1)^{i}{}_{k}C_{i}(k-i)^{n-1},$ (242)

and the Fermat’s little theorem that states $l^{n-1}\equiv 1\ {\rm mod}\ n$ if
$l$ is not divisible by a prime number $n$, one can show Eq. (241). Figure 22
shows the example of Eq. (241) for $n=7$.

Figure 22: Diagrammatic representation of Eq. (241) for $n=7$.

We remark that Eq. (240) holds for some odd composite numbers. For example,
$\\{a_{24}\langle n!\rangle\\}$ can be divided by 25. Also, for all the even
numbers, Eq. (240) does not hold because $\sum_{k=2}^{n-1}k!\times S_{n-1,k}\
{\rm mod}\ n$ is an even number mod $n$ and does not equal to $-1\ {\rm mod}\
n$.

###  9.2 Bell number

Bell number is a number counting how many number of ways to partition of a set
of $n$ elements. For example, $\\{a,b,c\\}$ can be partitioned into
$\\{a\\},\\{b\\},\\{c\\}$ or $\\{a\\},\\{b,c\\}$ or $\\{b\\},\\{c,a\\}$ or
$\\{c\\},\\{a,b\\}$ or $\\{a,b,c\\}$. Then, the third term of the Bell numbers
$B_{3}$ is 5. The $n$th term of Bell numbers is given by

$\displaystyle B_{n}=\sum_{k=0}^{n-1}{}_{n}C_{k}B_{k}.$ (243)

Then, one has

$\displaystyle\\{B_{n}\\}=1,1,2,5,15,52,203,877,4140,21147,\cdots.$ (244)

We remark that the factorials can be considered as dual of the Bell numbers in
terms of the Stirling number. First, the Bell numbers can be represented by
using the Stirling number of the second kind as follows.

$\displaystyle B_{n}=\sum_{k=0}^{n}S_{n,k}.$ (245)

Also, the unsigned Stirling number of the first kind $|s(n,k)|$ is a number
that counts the number of permutations of $n$ elements with $k$ disjoint
cycles. For example, $|s(3,2)|=3$ because there are three permutations of
three elements that fixed the one element: (1,2) or $(2,3)$ or $(3,1)$. Then,
the factorial can be represented as follows,

$\displaystyle n!=\sum_{k=0}^{n}|s_{n,k}|.$ (246)

Now, we construct the sequence dual of Bell numbers. First, the exponential
generating function of the $n$th Bell number is given by

$\displaystyle\sum_{n=0}^{\infty}\frac{B_{n}}{n!}x^{n}=e^{e^{x}-1}.$ (247)

Then, by differentiating the each side by $x$, one has

$\displaystyle\sum_{n=0}^{\infty}\frac{B_{n+1}}{n!}x^{n}=e^{x}e^{e^{x}-1}.$
(248)

Note that $e^{x}$ can be obtained by substituting $\alpha=1$ into $e^{\alpha
x}$. Then, we define the sequence dual of Bell numbers by using
$a_{1}[e^{\alpha x}]_{L}$ that can be obtained by substituting $n=1$ into
$\\{a_{n}[e^{\alpha x}]\\}_{L}$.

$\displaystyle\sum_{n=0}^{\infty}\frac{a_{n}\langle
B_{n}\rangle}{n!}\alpha^{n}$
$\displaystyle=\frac{1}{1-\alpha}e^{\left(\frac{1}{1-\alpha}-1\right)}=a_{1}[e^{\alpha
x}]_{L}\,e^{\left(a_{1}[e^{\alpha x}]_{L}-1\right)}.$ (249)

We remark that $\\{a_{n}\langle B_{n}\rangle\\}$ relates to $B_{n}$ through
the Stirling transform of the first kind corresponding to Eq. (238) [37].

######  Proposition 32 (Sequence dual of Bell numbers and Stirling transform).

$\displaystyle\\{a_{n}\langle B_{n}\rangle\\}$
$\displaystyle=\left\\{\sum_{k=0}^{n}|s(n,k)|\mathcal{S}_{1}B_{k}\right\\},$
(250) $\displaystyle=1,2,7,34,209,1546,13327,130922,1441729,17572114,\cdots.$
(251)

We will now focus on the following modular properties of Bell numbers
concerning prime numbers.

$\displaystyle B_{n+p}-B_{n}\equiv B_{n+1}\ \ ({\rm mod}\ p).$ (252)

Then, we have the following modular property for the sequence dual of Bell
numbers that should be regarded as sequence dual of the above theorem.

######  Theorem 5 (Sequence dual of modular property of Bell numbers).

For natural numbers $n$ and $m$,

$\displaystyle\\{a_{n+m}\langle B_{n}\rangle\\}-\\{a_{n}\langle
B_{n}\rangle\\}\equiv 0\ \ ({\rm mod}\ m).$ (253)

For example, $a_{7}\langle B_{n}\rangle-a_{6}\langle
B_{n}\rangle=130922-13327=1\times 117595$, $a_{7}\langle B_{n}\rangle-
a_{5}\langle B_{n}\rangle=130922-1546=2\times 64688$, $a_{7}\langle
B_{n}\rangle-a_{4}\langle B_{n}\rangle=130922-209=3\times 43571$,
$a_{7}\langle B_{n}\rangle-a_{3}\langle B_{n}\rangle=130922-34=4\times 32722$,
$a_{7}\langle B_{n}\rangle-a_{2}\langle B_{n}\rangle=130922-7=5\times 26183$,
$a_{7}\langle B_{n}\rangle-a_{1}\langle B_{n}\rangle=130922-2=6\times 21820$,
$a_{7}\langle B_{n}\rangle-a_{0}\langle B_{n}\rangle=130922-1=7\times 18703$.

The above theorem can be proved by using the recursion formula,

$\displaystyle a_{n}\langle B_{n}\rangle=2na_{n-1}\langle
B_{n}\rangle-(n-1)^{2}a_{n-2}\langle B_{n}\rangle,$ (254)

and induction method as follows.

$\displaystyle a_{n+1+m}\langle B_{n}\rangle$ $\displaystyle\equiv
2(n+1)a_{n}\langle B_{n}\rangle-n^{2}a_{n-1}\langle B_{n}\rangle\ {\rm mod}\
m\equiv a_{n+1}\langle B_{n}\rangle\ {\rm mod}\ m.$ (255)

Note that for the first two terms of $\\{a_{n}\langle B_{n}\rangle\\}$, the
following relations hold.

$\displaystyle a_{m}\langle B_{n}\rangle\equiv a_{0}\langle B_{n}\rangle\ {\rm
mod}\ m\equiv 1\ {\rm mod}\ m,$ (256) $\displaystyle a_{m+1}\langle
B_{n}\rangle-a_{1}\langle B_{n}\rangle\equiv a_{1}\langle B_{n}\rangle\ {\rm
mod}\ m\equiv 2\ {\rm mod}\ m.$ (257)

The first relation can be proved by the following formula for $\\{a_{n}\langle
B_{n}\rangle\\}$.

$\displaystyle a_{n}\langle B_{n}\rangle=\sum_{k=0}^{n}k!({}_{n}C_{k})^{2}.$
(258)

The second formula can be shown as follows.

$\displaystyle a_{1+m}\langle B_{n}\rangle$ $\displaystyle=2(1+m)a_{m}\langle
B_{n}\rangle-(1+m-1)^{2}a_{m-1}\langle B_{n}\rangle\equiv 2(1+m)\ {\rm mod}\
m$ $\displaystyle\equiv 2\ {\rm mod}\ m.$ (259)

##  10 Discussion

In this paper, we created a sequence version of calculus. First, we defined
equivalence, some fundamental operations, left/right differential and integral
for sequences. Then, we proposed sequence version of power function,
exponential function, hyperbolic function, and trigonometric function and
sequence versions of Maclaurin series for them. We found the sequence versions
of Maclaurin series for exponential function involve divergent series
including Grandi’s series. We remark that generally there are some candidates
of the sequence version of a given function. As for sequence versions of
trigonometric functions, first, two kinds of sequence versions of
trigonometric functions were proposed, depending on whether the left or right
differential was used. The right/left trigonometric functions are graphically
represented by the divergent/convergent spiral while the usual trigonometric
functions are graphically represented by a circle. Then, we found the sequence
version of binomial theorem and Euler’s identity. The sequence version of
Euler’s identity is composed of two equations determined by fourth term of
left/right sequence version of $e^{ix}$. Then, by combining the two sequence
versions of trigonometric functions, we proposed another sequence version of
trigonometric functions whose periodicity is 8. Furthermore, we proposed
sequence version of Maclaurin series for Fibonacci sequence, $(P,\
Q)$-Fibonacci sequence, and $k$-bonaaci sequence, and found similarities
between them and the sequence versions of exponential function. In addition,
we introduce sequence dual of factorial and Bell numbers. Then, the sequence
dual of the modular property of factorial concerning the prime numbers
(Wilson’s theorem) and of the Bell numbers concerning the prime numbers were
obtained. Finally, we remark a few points for future research. First,
probably, our theory relates systematically to the figurate number including
(centered) polygonal numbers, (centered) pyramidal numbers, or (centered)
Platonic numbers. Note that $\\{a_{n}[x^{2}/2]\\}_{R}$ is the triangle number.
Also, there are a few functions that we could not deal with in this paper such
as logarithmic function, inverse trigonometric/hyperbolic function, elliptic
integral, zeta function. Sequence version of these functions may lead to new
formulae for some sequences. Also, considering other frameworks of analogy
between sequence and analytics may lead to many new theorems of sequences and
analytics.

##  Appendix A List of sequence version of functions and sequence duals

Constant function:
$\\{a_{n}[a]\\}=a,a,a,a,a,a,a,a,a,a,a,a,\cdots$.
Power function:
$\\{a_{n}[x^{k}]\\}=0^{k},1^{k},2^{k},3^{k},4^{k},5^{k},6^{k},7^{k},8^{k},9^{k},10^{k},11^{k},\cdots$.
Exponential function (right):
$\\{a_{n}[e^{\alpha
x}]\\}_{R}=(1+\alpha)^{0},(1+\alpha)^{1},(1+\alpha)^{2},(1+\alpha)^{3},(1+\alpha)^{4},(1+\alpha)^{5},(1+\alpha)^{6},\\\
\qquad\qquad(1+\alpha)^{7},(1+\alpha)^{8},(1+\alpha)^{9},(1+\alpha)^{10},(1+\alpha)^{11},\cdots$.
Exponential function (left):
$\\{a_{n}[e^{\alpha
x}]\\}_{L}=(1-\alpha)^{0},(1-\alpha)^{-1},(1-\alpha)^{-2},(1-\alpha)^{-3},(1-\alpha)^{-4},(1-\alpha)^{-5},(1-\alpha)^{-6},\\\
\qquad\qquad(1-\alpha)^{-7},(1-\alpha)^{-8},(1-\alpha)^{-9},(1-\alpha)^{-10},(1-\alpha)^{-11},\cdots$.
Exponential function (natural):
$\\{a_{n}[e^{-x}]\\}^{\rm nat}=1,0,0,0,0,0,0,0,0,0,0,0,\cdots$.
Hyperbolic cosine function:
$\\{a_{n}[\cosh
x]\\}=1,\frac{5}{4},\frac{17}{8},\frac{65}{16},\frac{257}{32},\frac{1025}{64},\frac{4097}{128},\frac{16385}{256},\frac{65537}{512},\frac{262145}{1024},\frac{1048577}{2048},\frac{4194305}{4096},\cdots$.
Hyperbolic sine function:
$\\{a_{n}[\sinh
x]\\}=1,\frac{3}{4},\frac{15}{8},\frac{63}{16},\frac{255}{32},\frac{1023}{64},\frac{4095}{128},\frac{16383}{256},\frac{65535}{512},\frac{262143}{1024},\frac{1048575}{2048},\frac{4194303}{4096},\cdots$.
Hyperbolic cosine function (natural):
$\\{a_{n}[\cosh x]\\}^{\rm nat}=1,1,2,4,8,16,32,64,128,256,512,1024,\cdots$.
Hyperbolic sine function (natural):
$\\{a_{n}[\sinh x]\\}^{\rm nat}=0,1,2,4,8,16,32,64,128,256,512,1024,\cdots$.
Cosine function (right):
$\\{a_{n}[\cos x]\\}_{R}=1,1,0,-2,-4,-4,0,8,16,16,0,-32,\cdots$.
Sine function (right):
$\\{a_{n}[\sin x]\\}_{R}=0,1,2,2,0,-4,-8,-8,0,16,32,32,\cdots$.
Tangent function (right):
$\\{a_{n}[\tan x]\\}_{R}=0,1,\infty,-1,0,1,-\infty,-1,0,1,\infty,-1,\cdots$.
Cosine function (left):
$\\{a_{n}[\cos
x]\\}_{L}=1,\frac{1}{2},0,-\frac{1}{4},-\frac{1}{4},-\frac{1}{8},0,\frac{1}{16},\frac{1}{16},\frac{1}{32},0,-\frac{1}{64}\cdots$.
Sine function (left):
$\\{a_{n}[\sin
x]\\}_{L}=0,\frac{1}{2},\frac{1}{2},\frac{1}{4},0,-\frac{1}{8},-\frac{1}{8},-\frac{1}{16},0,\frac{1}{32},\frac{1}{32},\frac{1}{64},\cdots$.
Tangent function (left):
$\\{a_{n}[\tan x]\\}_{L}=0,1,\infty,-1,0,1,-\infty,-1,0,1,\infty,-1,\cdots$.
Cosine function (periodic)
$\\{a_{n}[\cos x]\\}^{\rm
per}=1,\frac{1}{\sqrt{2}},0,-\frac{1}{\sqrt{2}},-1,-\frac{1}{\sqrt{2}},0,\frac{1}{\sqrt{2}},1,\frac{1}{\sqrt{2}},0,-\frac{1}{\sqrt{2}},\cdots$.
Sine function (periodic)
$\\{a_{n}[\sin x]\\}^{\rm
per}=0,\frac{1}{\sqrt{2}},1,\frac{1}{\sqrt{2}},0,-\frac{1}{\sqrt{2}},-1,-\frac{1}{\sqrt{2}},0,\frac{1}{\sqrt{2}},1,\frac{1}{\sqrt{2}},\cdots$.
Factorial:
$\\{a_{n}\langle
n!\rangle\\}=2,2,6,26,150,1082,9366,94586,1091670,14174522,204495126,\\\
\qquad\qquad 3245265146,\cdots$.
Bell number:
$\\{a_{n}\langle
B_{n}\rangle\\}=1,2,7,34,209,1546,13327,130922,1441729,17572114,234662231,\\\
\qquad\qquad 3405357682,\cdots$.

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