Calculus of Sequence arXiv:2203.13676v2 [math.GM] 28 Mar ...

Differential and Integral Calculus of Sequence

Yusuke Imai ∗

Graduate School of Engineering Science, Osaka University,Toyonaka, Osaka 560-8531, Japan

April 5, 2022

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, exponentialfunction, hyperbolic function, trigonometric function, and also find sequenceversions of the Maclaurin series for them. The sequence versions of exponen-tial function involve divergent series including Grandi’s series. By using thisframework, we find a sequence version of the binomial theorem and Euler’sidentity. In addition, we design new formalisms of Fibonacci sequence and itsgeneralizations. Last, we propose a sequence dual of factorial and Bell num-ber, and find sequence dual of modular property of factorial concerning primenumber (Wilson’s theorem) and of Bell number concerning prime number.

Contents

1 Introduction 3

2 Definitions 102.1 Sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2 Equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.3 Fundamental operations . . . . . . . . . . . . . . . . . . . . . . . . . 102.4 Differential and Integral . . . . . . . . . . . . . . . . . . . . . . . . . 11

3 Fundamental sequence 13

4 Power function 15

∗CONTACT: [email protected]

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5 Exponential function 165.1 ex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165.2 e−x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205.3 eαx . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

5.3.1 {an[eαx]}R, α 6= −1 . . . . . . . . . . . . . . . . . . . . . . . . 245.3.2 {an[eαx]}L, α 6= 1 . . . . . . . . . . . . . . . . . . . . . . . . . 26

6 Hyperbolic function 28

7 Trigonometric function 31

8 Fibonacci sequences 358.1 Fibonacci sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358.2 (P, Q)-Fibonacci sequence . . . . . . . . . . . . . . . . . . . . . . . . 36

8.2.1 Example 1: Pell sequence (P = 2, Q = 1) . . . . . . . . . . . . 378.2.2 Example 2: Jacobsthal sequence (P = 1, Q = 2) . . . . . . . . 38

8.3 k-bonacci sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

9 Factorial and Bell number 409.1 Factorial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409.2 Bell number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

10 Discussion 44

A List of sequence version of functions and sequence duals 45

2

1 Introduction

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

1, 1, 1, 1, 1, 1, · · · . (1)

This sequence has the following three properties. First, all the numbers in the abovesequence are 1. Second, all the differences between the two consecutive terms are0. 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, wecan see that the above sequence is a geometric sequence. The following sequence isalso a fundamental sequence that is an arithmetic sequence whose common differenceis 1 and the initial term is given by 0.

0, 1, 2, 3, 4, 5, · · · . (2)

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

1, 2, 4, 8, 16, 32, · · · . (3)

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

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

0, 1, 1, 2, 3, 5, · · · . (4)

The most characteristic feature of the Fibonacci sequence is that the ratio of con-secutive two terms in the Fibonacci sequence approach to the golden ratio. TheFibonacci sequence was proposed in 1202 by Leonardo of Pisa [1], and it has beendiscovered that it characterizes many things in nature, such as bee populations, spi-ral patterns of poplar, willow, pear trees, beech, hazel, cherry, apple, elm, lime, andalmond [2].

In addition, there exist sequences, each of whose term is determined by its indexand the previous terms, except the first term. For example, each term of the follow-ing sequence is given by n times the previous term, except the first term if the firstindex is set to 0.

1, 1, 2, 6, 24, 120, · · · . (5)

The nth term in the above sequence equals to factorial of n that represents thenumber of permutations for n elements. The (n, k)th unsigned Stirling numbers ofthe first kind represents the number of permutations for n elements with k disjoint

3

cyclic permutations. Then, sum of the (n, k)th unsigned Stirling number of the firstkind from k = 0 to n equals to the nth factorial [3] (Fig. 1(a)).

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

1, 1, 2, 5, 15, 52, · · · . (6)

The nth 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 nelements by k disjoint groups. Then, sum of the (n, k)th Stirling number of thesecond kind from k = 0 to n equals to the nth Bell number. While the (n, k)thunsigned Stirling numbers of the first kind can also be defined as coefficients in alinear combination of xk for k = 0, · · · , n that equals to the rising factorial, x(x +1) · · · (x+n−1), the (n, k)th Stirling number of the second kind can also be definedas coefficients in a linear combination of the falling factorial x(x− 1) · · · (x− k+ 1)for k = 0, · · · , n that equals to xn [3] (Fig. 1(b)). Furthermore, both the factorialsand the Bell numbers have the modular property concerning prime numbers. Thefollowing theorem is called Wilson’s theorem [7, 8, 9] and states that the (p− 1)thfactorial equals to −1 modulo p if and only if p is a prime number.

Theorem.

(p− 1)! ≡ −1 (mod p). (7)

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

Theorem.

Bn+p ≡ Bn +Bn+1 (mod p). (8)

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

f(x) = 1. (9)

This function has the following three properties. First, all the values the abovefunction 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 afundamental real function that is called identity function and that satisfies f(x +1)− f(x) = 1.

f(x) = x. (10)

The following theorem is called binomial theorem [12].

Theorem.

(1 + x)n =n∑

k=0

nCk xk. (11)

4

k = 1

k = 2

k = 3

(a) 3! = 6

1

2 3

1

3 2

3

2 1

2

1 3

2

3 1

3

1 2

(b) B3 = 5

31 2

31 2 12 3 23 1

31 2

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

If x = 1, the binomial theorem can be visualized by Pascal’s triangle (Fig. 2).The following real function is a fundamental real function that is called exponen-

tial function and that satisfies f(x+ 1)/f(x) = e where e is the Napier’s constant.

f(x) = ex. (12)

The exponential function is often used to represent an explosive increase of quan-tity such as the number of infected persons [13]. The exponential function is alsocharacterized by invariance for differential as follows.

d

dxex = ex. (13)

The hyperbolic functions, sinhx and coshx, are determined by the exponentialfunction as follows.

coshx =ex + e−x

2, (14)

sinhx =ex − e−x

2. (15)

Perhaps the most famous application of hyperbolic functions is the four-dimensionalrotation in special relativity [14]. The hyperbolic functions satisfy the followingproperty for differential.

d

dxcoshx = sinhx, (16)

d

dxsinhx = coshx. (17)

5

1

1 2 1

1 4 6 4 1

1 3 3 1

1 5 10 10 5 1

Figure 2: Pascal’s triangle.

Also, the trigonometric functions, cosx and sinx, are determined by the exponentialfunction as follows.

cosx =eix + e−ix

2, (18)

sinx =eix − e−ix

2i. (19)

They are used to represent rotation in Euclidean space [15], and cos x and sinx havethe following periodic property.

cos(x+ 2π) = cos x, (20)

sin(x+ 2π) = sin x. (21)

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

d

dxcosx = − sinx, (22)

d

dxsinx = cosx. (23)

One can see that the differential rule for the trigonometric functions is similar toone of the hyperbolic functions. The following relations also indicate the similaritybetween the trigonometric functions and the hyperbolic functions.

cos2 x+ sin2 x = 1, (24)

cosh2 x− sinh2 x = 1, (25)

cos2 x− sin2 x = cos 2x, (26)

cosh2 x+ sinh2 x = cosh 2x, (27)

2 cosx sinx = sin 2x, (28)

2 coshx sinhx = sinh 2x. (29)

6

The exponential function, the hyperbolic functions, and the trigonometric functionshave the Maclaurin series, i.e. can be represented by the series of 1, x, x2, · · · withcoefficients.

ex =∞∑

n=0

xn

n!, (30)

coshx =∞∑

n=0

x2n

(2n)!, (31)

sinhx =∞∑

n=0

x2n+1

(2n+ 1)!, (32)

cosx =∞∑

n=0

(−1)nx2n

(2n)!, (33)

sinx =∞∑

n=0

(−1)nx2n+1

(2n+ 1)!. (34)

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

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

eix = cosx+ i sinx. (35)

The following equality is called Euler’s identity [16] and would be the most funda-mental and beautiful theorem in mathematics because it connects various mathe-matical concepts, the e (Napier’s constant), 0 (additive identity), 1 (multiplicativeidentity), i (imaginary unit), π (circle ratio).

Theorem.

eiπ + 1 = 0. (36)

Figure 3(a) shows Euler’s identity (eiπ + 1 = 0) and Fig. 3(b) shows e2iπ = 1,and Fig. 3(c) shows geometric representation of eiπ × eiπ = 1.

ei⇡

0

+1Re

Im(a)

Re

Im

e2i⇡

(b) (c)

Figure 3: Diagrammatic representation of (a) Euler’s identity (eiπ + 1 = 0) and (b)e2iπ = 1, and (c) geometric representation of eiπ× eiπ = 1 (area of the square equalsto 1).

7

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

Γ(z) =

∫ ∞

0

tz−1e−tdt. (37)

The gamma function satisfies the following relation

Γ(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 repre-

sented as follows [4].

eex

/e =∞∑

n=0

Bnxn

n!. (39)

Then, the above function is called the exponential generating function of Bn. Notethat the exponential generating function of 1, 1, 1, · · · is given by the exponentialfunction.

This paper investigates relations among the sequences and functions shown aboveand 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 solvediscrete differential equations. In this paper, by inheriting the concepts proposed inRef. [18] with some retouching, we build a sequence version of calculus. Because ofthe difference between the discreteness and the continuity, it is difficult to constructa 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 discreteversion of some elementary functions: exponential function, logarithmic function,and trigonometric function. The note concludes with the sentences after dealing withthe 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 versionof the trigonometric functions. Could not a more appropriate function be found?What do the readers think?”. Because, in this paper, we propose three kinds ofsequence 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 trigonometricfunctions are entangled deeply each other to result in various formulae that can beregarded as sequence version of formulae of the trigonometric functions, we wishthis paper can be one of the answers to that question.

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

8

In Sec. 3, we define the sequence version of fundamental real functions suchas additive identity (f(x) = 0), multiplicative identity (f(x) = 1), and generallyconstant function (f(x) = a) where a is a real number. We also introduce thegeneral 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) = xa) where a is a real number. Also, we showthe law of exponent holds for the sequence version of the power function. Then, weobtain a direct analogy of the binomial theorem (Eq. (11)). This can be proved byusing the usual binomial theorem.

In Sec. 5, we first define the sequence version of exponential function by focusingon the property of the exponential function for differential (Eq. (13)). Then, weobtain 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 theright/left integral of the sequence version of xn relates to xn+1 through the Euleriannumber. Next, we define two kinds of sequence version of the negative exponentialfunction depending on whether the left differential or right differential is used. Last,we define two kinds of sequence versions of the general exponential function whoseexponent is given by a general real number a.

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

In Sec. 7, we first define two kinds of sequence version of the sine function andcosine function by using Eq. (35) and the two kinds of sequence version of thegeneral exponential function defined in Sec. 5. Each of them satisfies the sequenceversion of Eq. (24), Eq. (26), and Eq. (28). Then, we obtain sequence version ofEuler’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) functionwhose periodicity is 8.

In Sec. 8, we propose the sequence version of the Maclaurin series for the Fi-bonacci 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 simi-larities between those generalizations of the Fibonacci sequence and the sequenceversion of exponential function from the viewpoint of the sequence version of theMaclaurin series.

In Sec. 9, we introduce sequence dual of factorial and of Bell numbers by focusingon the functions shown in Eq. (37) and Eq. (39), and we find that sequence dualof 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 sequencedual of the factorial (Bell numbers) inherits partially the modular property of thefactorial (Bell numbers) for the prime numbers shown in Eq. (7) (Eq. (8)).

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

9

2 Definitions

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

2.1 Sequence

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

{an} = a0, a1, a2, · · · . (40)

2.2 Equivalence

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

Definition 1 (Equivalence). For any sequences {an} and {bn}, {an} = {bn} if andonly if for any non-negative integer n,

an = bn. (41)

2.3 Fundamental operations

In this subsection, we define fundamental operations for a sequence or two sequences.First, we define shift operator Sk as follows.

Definition 2 (Shift operator). For any sequence {an} and non-negative integer k,

Skan = an+k, Sk{an} = {an+k}. (42)

The sum (+), subtraction (−), multiplication (×), and division (/) of two se-quences are naturally defined as follows.

Definition 3 (Sum, Subtraction, Multiplication, Division). For any sequences {an}and {bn},

{an}+ {bn} = {an + bn}, (43)

{an} − {bn} = {an − bn}, (44)

{an} × {bn} = {an × bn}. (45)

For any sequence {an} and {bn} that does not include 0,

{an}/{bn} = {an/bn}. (46)

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

Definition 4 (Scalar addition and multiplication). For any sequence {an} and areal number α,

{an + α} = {an}+ α, (47)

{αan} = α{an}. (48)

10

Inverse of a given sequence {an} that does not include 0 is defined as follows.

Definition 5 (Inverse). We represent inverse of {an} that does not include 0 by{an}−1 and define {an}−1 as

{an}−1 = a−10 , a−11 , · · · . (49)

Also, we define insertion operator Iα for a real number α as follows.

Definition 6 (Insertion).

Iα{an} = α, a0, a1, · · · . (50)

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

S1Iα{an} = {an}. (51)

2.4 Differential and Integral

In this subsection, we define differential and integral of sequence. First, we proposetwo definitions of the differential of sequence: right differential and left differential.The right differential DR{an}R of a sequence {an} is defined as follows.

Definition 7 (Right Differential). For any sequence {an},

DR{an} = {an+1 − an}. (52)

We also use the left differential of DL{an}L of a sequence {an}.

Definition 8 (Left differential). For any sequence {an},

DL{an} = {an − an−1}. (53)

To make the above definition well defined, a−1 is needed. We add an appropriatea−1 to {an} if necessary.

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

DR = S1DL. (54)

We also define the right integral IαR{an} of a sequence {an} is defined as follows.

Definition 9 (Right Integral). For any sequence {an} and an integral constant thatis a real number, α,

IαR{an} =

{n∑

k=0

ak

}+ α. (55)

In addition, we define the left integral IαL{an} of a sequence {an} as follows.

11

Definition 10 (Left Integral). For any sequences {an} and an integral constantthat is a real number, α,

IαL{an} =

{n−1∑

k=0

ak

}+ α, (56)

where we assumed∑−1

k=0 ak = 0.

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

IαR = S1IαL . (57)

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

DLIαR{an} =

{n∑

k=0

ak −n−1∑

k=0

ak

}= {an}, (58)

IαRDL{an} =

{n∑

k=0

(ak − ak−1)}

+ α = {an} − a−1 + α, (59)

DRIαL{an} =

{n∑

k=1

ak −n−1∑

k=1

ak

}= {an}, (60)

IαLDR{an} =

{n−1∑

k=0

(ak+1 − ak)}

+ α = {an − a0}+ α = {an} − a0 + α. (61)

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

DRIαR{an} =

{n+1∑

k=0

ak −n∑

k=0

ak

}= {an+1} = S1{an}, (62)

IαRDR{an} =

{n∑

k=0

(ak+1 − ak)}

= {an+1 − a0}+ α = S1{an} − a0 + α, (63)

S1DLIαL{an} =

{n∑

k=0

ak −n−1∑

k=0

ak

}= {an}, (64)

S1IαLDL{an} =

{n∑

k=0

(ak − ak−1)}

= {an − a0}+ α = {an} − a0 + α. (65)

In Fig. 4, we show how the left/right differential and integral act on a sequence,{an} = 1, 3, 2, 5, 6, 4, · · · and a−1 = 0.

12

1 3 2 5 6 4 ・・・

2 -1 3 1 -2 ・・・

0

n=-1 n=0 n=1 n=2 n=3 n=4 n=5

DR{an}

{an}

DL{an} 2 -1 3 1 -21 ・・・

1

0

4 6 11 17 21

1 4 6 11 17 21

・・・

・・・

I0R{an}

I0L{an}

Figure 4: The left/right differential and integral of a sequence {an} =1, 3, 2, 5, 6, 4, · · · and a−1 = 0.

3 Fundamental sequence

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

Definition 11 (Sequence version of 0).

{an[0]} = {0} = 0, 0, 0, · · · . (66)

Then, any sequence {an} satisfies the relations

{an}+ {an[0]} = {an[0]}+ {an} = {an}, (67)

{an} − {an[0]} = {an}, (68)

{an[0]} − {an} = {−an} = −{an}. (69)

In addition, the multiplicative identity sequence {an[1]} should be defined as follows.

Definition 12 (Sequence version of 1).

{an[1]} = {1} = 1, 1, 1, · · · . (70)

Then, for any sequence {an},

{an} × {an[1]} = {an[1]} × {an} = {an}, (71)

{an}/{an[1]} = {an}, (72)

and for any sequence {an} that does not include 0,

{an[1]}/{an} = {an}−1. (73)

Generally, sequence version of constant function {an[a]} where a is a real number isgiven by

13

Definition 13 (Sequence version of a).

{an[a]} = {a} = a, a, a, · · · . (74)

The sequence version of constant function {an[a]} satisfies the following relationsfor any sequence {an}.

{an}+ {an[a]} = {an[a]}+ {an} = {an + a} = {an}+ a, (75)

{an} − {an[a]} = {an − a} = {an} − a, (76)

{an[a]} − {an} = {a− an} = a− {an}, (77)

{an} × {an[a]} = {an[a]} × {an} = {aan} = a{an}, (78)

{an}/{an[a]} = {an/a} = {an}/a, (79)

{an[a]}/{an} = {a/an} = a{an}−1, (80)

where we assumed a 6= 0 in Eq. (79) and an 6= 0 for any non-negative integer nin Eq. (80). In this paper, we use the following notation to represent a sequenceversion of a given function f(x),

Definition 14.

{an[f(x)]} = a0[f(x)], a1[f(x)], · · · . (81)

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4 Power function

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

Definition 15 (Sequence version of x).

{an[x]} = {n} = 0, 1, 2, · · · . (82)

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

Definition 16 (Sequence version of xa). For a real number, a,

{an[xa]} = {na} = 0a, 1a, 2a, · · · . (83)

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

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

{an[xa+b]} = {na+b} = 0a+b, 1a+b, 2a+b, · · ·= 0a × 0b, 1a × 1b, 2a × 2b, · · · = {an[xa]} × {an[xb]}, (84)

{an[xa−b]} = {na−b} = 0a−b, 1a−b, 2a−b, · · ·

=0a

0b,

1a

1b,

2a

2b, · · · = {an[xa]}/{an[xb]}, (85)

{an[xab]} = {nab} = 0ab, 1ab, 2ab, · · ·= (0a)b, (1a)b, (2a)b, · · · = {an[xa]}b. (86)

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

(1 + x)n =n∑

k=0

nCkxk. (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 numbern,

({an[1]}+ {an[x]})n= nC0{an[1]}n{an[x]}0 + nC1{an[1]}n−1{an[x]}1 + · · ·+ nCn{an[1]}0{an[x]}n

(88)

=n∑

k=0

nCk{an[1]}n−k{an[x]}k. (89)

15

・・・

++

+

・・・

・・・

・・・

・・・

1n ⇥ nCn

21 ⇥ nC1

22 ⇥ nC2

2n ⇥ nCn

10 ⇥ nC0

11 ⇥ nC1

12 ⇥ nC2

・・・

・・・

20 ⇥ nC0

・・・

2n1n 3n

0n ⇥ nCn

01 ⇥ nC1

02 ⇥ nC2

00 ⇥ nC0

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 focusingon differential property of the exponential function.

5.1 ex

The exponential function satisfies the following formula for differential.

d

dxex = ex. (90)

Then, we find a sequence {an[ex]}R that satisfies the following formula1.

DR{an[ex]}R = {an[ex]}R. (91)

The {an[ex]}R can be uniquely determined except the initial term as follows.

an+1 − an = an, (92)

an+1 = 2an. (93)

Then, we define {an[ex]}R as follows because e0 = 1.

Definition 17 (Right sequence version of exponential function).

{an[ex]}R = {2n} = 1, 2, 4, · · · . (94)

Figure 6 shows difference between the usual exponential function (ex) and theright sequence version of the exponential function ({an[ex]}R).

1In Remark 2, we define {an[ex]}L by using DL

16

0

20

40

60

80

100

120

140

160

0 1 2 3 4 5

{an [ex]}Rex

x

y

0 1 32 4 5020

40

6080

100

120

140160

Figure 6: Difference between ex and {an[ex]}R.

We remark that the difference between d/dx and DR is manifested by the dif-ference between the e and the 2. Then, we show sequence version of the Maclaurinseries for ex. By using the Maclaurin series, the ex is expressed as follows.

ex =∞∑

n=0

xn

n!=

1

0!1 +

1

1!x+

1

2!x2 +

1

3!x3 + · · ·

= 1 + x+

∫ x

0

x+

∫ x

0

∫ x

0

x+

∫ x

0

∫ x

0

∫ x

0

x+ · · · . (95)

Because we have already defined the sequence version of 1, x, ex, and the left integralof a sequence, then, we have the following left sequence version of the Maclaurinseries for {an[ex]}R as follows.

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

{an[ex]}R = {an[1]}+ {an[x]}+ I0L{an[x]}+ I0LI0L{an[x]}+ I0LI0LI0L{an[x]}+ · · · .(96)

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

I0L{an[x]} = 0, 0, 1, 3, 6, 10, · · · , (97)

I0LI0L{an[x]} = 0, 0, 0, 1, 4, 10, 20, · · · , (98)

I0LI0LI0L{an[x]} = 0, 0, 0, 0, 1, 5, 15, 35, · · · , (99)

etc. Then, we introduce the following definition.

Definition 18 (Left sequence version of xn/n!). For an integer k ≥ 2,

{an[xk/k!]}L = I0L · · · I0L︸ ︷︷ ︸k−1 times

{an[x]}. (100)

17

We remark {an[x2]} 6= {an[x2/2]}L+{an[x2/2]}L. Instead, the following formulaholds.

{an[x2]} = {an[x2/2]}L + S1{an[x2/2]}L. (101)

Generally, we obtain the following proposition.

Proposition 3 (Relation between left sequence version of xn/n! and sequence ver-sion of xn). For a natural number n,

{an[xn]} =n−1∑

k=0

A(n, k)Sk{an[xn/n!]}L, (102)

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

A(n, k) =k+1∑

l=0

(−1)l(n+ 1l

)(k + 1− l)n. (103)

The Eulerian number represents the number of permutations for n elements withk raises. For example, A(3, 2) = 1 because there exist only one permutation for 3elements 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 versionof exponential function 2).

{an[ex]}R =∞∑

n=0

{an[xn/n!]}L. (104)

By using the right integral of a sequence, we have the following right sequenceversion of the Maclaurin series for {an[ex]}R as follows.

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

{an[ex]}R = {an[1]}+1

2{an[x]}+

1

4I0R{an[x]}

+1

8I0RI0R{an[x]}+

1

16I0RI0RI0R{an[x]}+ · · · . (105)

Also, we have

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

{an[ex]}R =∞∑

n=0

1

2n{an[xn/n!]}R. (106)

18

Here, we defined {an[xn/n!]}R as follows.

Definition 19 (Right sequence version of xn/n!). For an integer k ≥ 2,

{an[xk/k!]}R = I0R · · · I0R︸ ︷︷ ︸k−1 times

{an[x]} = Sk−1{an[xk/k!]}L. (107)

For example, we have

I0R{an[x]} = 0, 1, 3, 6, 10, · · · , (108)

I0RI0R{an[x]} = 0, 1, 4, 10, 20, · · · , (109)

I0RI0RI0R{an[x]} = 0, 1, 5, 15, 35, · · · . (110)

Also, the first three terms of the right sequence version of the Maclaurin series for{an[ex]}R are represented as follows.

1 + 0 + 0 + · · · = 1, (111)

1 +1

2+

1

4+ · · · =

∞∑

n=0

1

2n= 2, (112)

1 +2

2+

3

4+ · · · =

∞∑

n=0

n+ 1

2n= 4. (113)

In addition, we obtain the following relation between the right sequence version ofxn/n! and sequence version of xn.

Proposition 7 (Relation between right sequence version of xn/n! and sequenceversion of xn). For a natural number n,

{an[xn]} =n−1∑

k=0

A(n, k)In−1−k0 {an[xn/n!]}R, (114)

Note that the factors 1/2, 1/22 etc. are needed to construct the right sequenceversion of Maclaurin series for {an[ex]}R, and replacement of the

∫ x0

by the I0L doesnot directly lead to the right sequence version of Maclaurin series for {an[ex]}R. Inthis paper, the factors 1/2, 1/22 etc. frequently play the role of bridge between theworld 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 and8.

Remark 2. We used DR to define {an[ex]}R in Eq. (91). The {an[ex]}L shouldsatisfy the following properties for any non-negative integer n.

an − an−1 = an, (115)

an−1 = 0. (116)

Then,

{an[ex]}L = 0, 0, · · · . (117)

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

19

1 1 1 1 1 1 ・・・

0 1 2 3 4 5 ・・・

・・・

0 0 1 3 6 10 ・・・

0 0 0 1 4 10 ・・・

0 0 0 0 1 5 ・・・

0 0 0 0 0 1 ・・・

+++++

1 2 4 8 16 32

・・・

・・・

・・・

・・・

・・・

Pascal’s triangle{an[1]}{an[x]}

・・・ {an[ex]}R

・・・

{an[x2/2]}L

{an[x3/6]}L

{an[x4/24]}L

{an[x5/120]}L

Figure 7: Diagrammatic representation of left sequence version of Maclaurin seriesfor {an[ex]}R that is equivalent to Pascal’s triangle.

5.2 e−x

Next, we define the sequence version of e−x. The e−x satisfies the following formula.

d

dxe−x = −e−x. (118)

Then, we find a sequence {an[e−x]}L that satisfies the following formula.

DL{an[ex]}L = −{an[ex]}L. (119)

The {an[e−x]}L can uniquely be determined except the initial term as follows.

an − an−1 = −an, (120)

2an = an−1. (121)

Then, we define {an[e−x]}L = a0[e−x]L, a1[e

−x]L, · · · as follows because e−0 = 1.

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

{an[e−x]}L = {2−n} = 1,1

2,

1

4, · · · , (122)

where we assumed a−1[e−x]L = 2.

By combining the right sequence version of exponential function and the leftsequence version of negative exponential function, we have the following inverserelation.

20

{an [ex]} = 1, 2, 4, 8, ・・・{an [1]}

{an [x]}

{an [x2/2]}L

{an [x3/6]}L

Figure 8: Diagrammatic representation of left sequence version of Maclaurin seriesfor {an[ex]}R by using blocks.

0 1 4 10 20 35 ・・・

0 1 4 10 2000 1 4 10 2000 1 4 10 2000 1 4 10 200

0 0 1 4 100

+

+

++

+

0 {an[x3]}1 8 27 64 125

・・・

・・・

・・・

・・・

・・・

・・・

A(3, 0) = 1

A(3, 1) = 4

A(3, 2) = 1{an[x3/6]}L = I20{an[x3/6]}R

S1{an[x3/6]}L = I0{an[x3/6]}R

S1{an[x3/6]}L = I0{an[x3/6]}R

S1{an[x3/6]}L = I0{an[x3/6]}R

S1{an[x3/6]}L = I0{an[x3/6]}R

S2{an[x3/6]}L = {an[x3/6]}R

Figure 9: Diagrammatic representation of the relation among {an[xn]},{an[xn/n!]}L, and {an[xn/n!]}R with the Eulerian number A(n, k) for n = 3.

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

{an[(ex)]}−1R = {an[e−x]}L. (123)

This is analogous to the following relation.

(ex)−1 = e−x. (124)

As with the left sequence version of Maclaurin series for {an[ex]}R, the left se-quence version of Maclaurin series for {an[e−x]}L can be obtained. First, the Maclau-rin series for the e−x is expressed as follows.

e−x =∞∑

n=0

(−1)nxn

n!=

1

0!1− 1

1!x+

1

2!x2 − 1

3!x3 + · · · (125)

= 1− x+

∫ x

0

x−∫ x

0

∫ x

0

x+

∫ x

0

∫ x

0

∫ x

0

x− · · · . (126)

21

Then, the left sequence version of Maclaurin series for {an[e−x]}L is obtained asfollows.

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

{an[e−x]}L = {an[1]} − 1

2{an[x]}+

1

22I0L{an[x]}

− 1

23I0LI0L{an[x]}+

1

24I0LI0LI0L{an[x]} − · · · . (127)

By using {an[xn/n!]}L, we can rewrite Prop. 9 as follows.

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

{an[e−x]}L =∞∑

n=0

(−1)n

2n{an[xn/n!]}L. (128)

Also, by using the left integral, we obtain the following proposition of the rightsequence version of Maclaurin series for {an[e−x]}L.

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

{an[e−x]}L = {an[1]} − {an[x]}+1

22I0L{an[x]} − I0LI0L{an[x]}

+ I0LI0LI0L{an[x]} − · · · . (129)

By using {an[xn/n!]}R, we can rewrite Prop. 11 as follows.

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

{an[e−x]}L =∞∑

n=0

(−1)n{an[xn/n!]}R. (130)

We remark that the sums in Props. 11 and 12 should not be interpreted by theusual sum but the Abel sum [23]. For example, the first term and second term ofthe right sequence version of Maclaurin series for {an[e−x]}L are given as follows.

1− 1 + 1− 1 + 1− 1 + · · · = 1

2= a1[x

n/n!]R, (131)

1− 2 + 3− 4 + 5− 6 · · · = 1

4= a2[x

n/n!]R. (132)

22

The first one is called Grandi’s series [24], and would be one of the most famousdivergent series. The right sequence version of Maclaurin series for all the terms of{an[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 divergentseries. Note that the nth term of {an[e−x]}L can be represented as follows.

an[e−x]L =1

(1 + x)n

∣∣∣∣x=1

. (133)

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

1

(1 + x)n= 1− nx+

n(n+ 1)

2x2 − n(n+ 1)(n+ 2)

6x3 + · · ·

= 1 +∞∑

k=1

n(n+ 1) · · · (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 ver-sion of Maclaurin series for {an[e−x]}L (left sequence version of Maclaurin series for{an[e−x]}L).

1 1 1 1 1 1 ・・・

0 -1/2 -2/2 -3/2 -4/2 -5/2 ・・・

・・・

0 0 1/4 3/4 6/4 10/4 ・・・

0 0 0 -1/8 -4/8 -10/8 ・・・0 0 0 0 1/16 5/16 ・・・0 0 0 0 0 -1/32 ・・・

+

+

+

1 1/2 1/4 1/8 1/16 1/32

・・・

・・・

・・・

・・・

・・・

{an[1]}

・・・

�{an[x]}/2

+

+

{an[e�x]}L

・・・

{an[x2/2]}L/22

�{an[x3/6]}L/23

{an[x4/24]}L/24

�{an[x5/120]}L/25

Figure 10: Diagrammatic representation of right sequence version of Maclaurin seriesfor left sequence version of negative exponential function

Remark 5. The {an[e−x]}R should satisfy the following properties for any non-negative integer n.

an+1 − an = −an, (135)

an+1 = 0. (136)

23

1 1 1 1 1 1 ・・・

0 ・・・

・・・

0 ・・・

0 ・・・

0 ・・・

0 ・・・

+

+

+

1 1/2 1/4 1/8 1/16 1/32

・・・

・・・

・・・

・・・

・・・

{an[1]}

・・・

+

+

{an[e�x]}L

・・・

-1

1-11-1

-2 -3 -4 -53 6 10 15-4 -10 -20 -355 15 35 70-6 -21 -56 -126

{an[x2/2]}R

�{an[x]}R

�{an[x3/6]}R

{an[x4/24]}R

�{an[x5/120]}R

Figure 11: Diagrammatic representation of left sequence version of Maclaurin seriesfor left sequence version of negative exponential function

Then,

{an[e−x]}R = a, 0, 0, 0, · · · , (137)

where a is a real number.

Remark 6. The sequence {an[e−x]}nat naturally corresponding to e−x from theviewpoint of Maclaurin series is given by

{an[e−x]}nat = {an[1]} − {an[x]}+ I0L{an[x]} − I0LI0L{an[x]}+ I0LI0LI0L{an[x]} − · · · = {(1− 1)n} = 1, 0, 0, · · · , (138)

where we assumed 00 = 1.

5.3 eαx

In this subsection, we generalize the result obtained in the previous two subsections,and define sequence version of eαx for any real number α.

5.3.1 {an[eαx]}R, α 6= −1

A sequence satisfying the following relation

DR{an[eαx]}R = α{an[eαx]}R. (139)

is given by

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

{an[eαx]}R = 1, (1 + α), (1 + α)2, · · · . (140)

24

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

e(a+b)x = eaxebx. (141)

However, the law of exponent for {an[eαx]}R itself does not hold and instead, thefollowing relation holds.

Proposition 13 (Law of exponent for right sequence version of general exponentialfunction). For real numbers α and β,

{an[e(α+β+αβ)x]}R = {an[eαx]}R{an[eβx]}R. (142)

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

Proposition 14 (Left sequence version of Maclaurin series for right sequence versionof general exponential function 1). The sequence version of Maclaurin series for{an[eαx]}R is given by

{an[eαx]}R = {an[1]}+ α{an[x]}+ α2I0L{an[x]}+ α3I0LI0L{an[x]}+ α4I0LI0LI0L{an[x]}+ · · · . (143)

Proposition 15 (Left sequence version of Maclaurin series for right sequence versionof general exponential function 2). The sequence version of Maclaurin series for{an[eαx]}R is given by

{an[eαx]}R =∞∑

n=0

αn{an[xn/n!]}L. (144)

Proposition 16 (Right sequence version of Maclaurin series for right sequenceversion of general exponential function 1). The sequence version of Maclaurin seriesfor {an[eαx]}R is given by

{an[eαx]}R = {an[1]}+α

1 + α{an[x]}+

(α

1 + α

)2

I0R{an[x]}+

(α

1 + α

)3

I0RI0R{an[x]}

+

(α

1 + α

)4

I0RI0RI0R{an[x]}+ · · · . (145)

Proposition 17 (Right sequence version of Maclaurin series for right sequenceversion of general exponential function 2). The sequence version of Maclaurin seriesfor {an[eαx]}R is given by

{an[eαx]}R =∞∑

n=0

(α

1 + α

)n{an[xn/n!]}R. (146)

The right sequence version of Maclaurin series should be interpreted throughBorel summation [23]. The following formula usually holds if |α| < 1.

1 +α

1 + α+

(α

1 + α

)2

+ · · · = 1 + α. (147)

25

However, if the r.h.s. is interpreted by the Borel summation, for all α 6= −1, thedivergent series appeared in the right sequence version of Maclaurin series can havevalue of real numbers. For example, the sequence version of Maclaurin series for thea1[e

− 23x]R is given by

1− 2 + 4− 8 + 16− 32 + · · · = 1

3= a1[e

− 23x]R. (148)

5.3.2 {an[eαx]}L, α 6= 1

A sequence satisfying the following relation

DL{an[eαx]}L = α{an[eαx]}L, (149)

is given by

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

{an[eαx]}L = 1, (1− α)−1, (1− α)−2, · · · . (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 exponentialfunction). For real numbers α and β,

{an[e(α+β−αβ)x]}L = {an[eαx]}L{an[eβx]}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 func-tion). For a real number α that is not equals to −1,

{an[eαx]}−1R = {an[e−αx]}L. (152)

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

Proposition 20 (Left sequence version of Maclaurin series for left sequence versionof general exponential function 1). The sequence version of Maclaurin series for{an[eαx]}L is given by

{an[eαx]}L = {an[1]}+α

1− α{an[x]}+

(α

1− α

)2

I0L{an[x]}

+

(α

1− α

)3

I0LI0L{an[x]}+

(α + 1

α

)4

I0LI0LI0L{an[x]}+ · · · .(153)

Proposition 21 (Left sequence version of Maclaurin series for left sequence versionof general exponential function 2). The sequence version of Maclaurin series for{an[eαx]}R is given by

{an[eαx]}L =∞∑

n=0

(α

1− α

)n{an[xn/n!]}L. (154)

26

Proposition 22 (Right sequence version of Maclaurin series for left sequence versionof general exponential function 1). The sequence version of Maclaurin series for{an[eαx]}R is given by

{an[eαx]}L = {an[1]}+ α{an[x]}+ α2I0R{an[x]}+ α3I0RI0R{an[x]}+ α4I0RI0RI0R{an[x]}+ · · · . (155)

Proposition 23 (Right sequence version of Maclaurin series for left sequence versionof general exponential function 2). The sequence version of Maclaurin series for{an[eαx]}R is given by

{an[eαx]}L =∞∑

n=0

αn{an[xn/n!]}R. (156)

27

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 asfollows.

coshx =ex + e−x

2=∞∑

n=0

x2n

(2n)!, (157)

sinhx =ex − e−x

2=∞∑

n=0

x2n+1

(2n+ 1)!, (158)

and we have already defined the sequence version of the ex in Eq. (94) and e−x inEq. (122), then, we can obtain sequence version of the hyperbolic cosine functionand hyperbolic sine function as follows.

Definition 23 (Sequence version of hyperbolic function).

{an[coshx]} =1

2[{an[[ex]]}R + {an[e−x]}L] =

1

2{2n + 2−n} = 1,

5

4,17

8,65

16, · · · ,

(159)

{an[sinhx]} =1

2[{an[[ex]]}R − {an[e−x]}L] =

1

2{2n − 2−n} = 0,

3

4,15

8,63

16, · · · .

(160)

Figure 12(a) shows difference between the usual hyperbolic cosine function (coshx)and the sequence version of the hyperbolic cosine function ({an[coshx]}) and Fig. 12(b)shows difference between the usual hyperbolic sine function (sinhx) and the sequenceversion of the hyperbolic sine function ({an[sinhx]}).

0

10

20

30

40

50

60

70

80

0 1 2 3 4 5

(a) (b) (c){an [cosh x]}cosh x

x

y

0 1 32 4 501020

30

40

50

60

7080

0

10

20

30

40

50

60

70

80

0 1 2 3 4 5

{an [sinh x]} sinh x

x

y

0 1 32 4 501020

30

40

50

60

7080

0

0.05

0.1

0.15

0.2

0.25

0 1 2 3 4 5

x

y

{an [cosh x]} - an [cosh x]}nat

= {an [sinh x]}nat - an [sinh x]}

32 4 500 1

0.05

0.1

0.15

0.2

0.25

Figure 12: Difference between (a) coshx and {an[coshx]}, (b) sinhx and{an[sinhx]}, (c) {an[coshx]} ({an[sinhx]}nat) and {an[coshx]}nat ({an[sinhx]}).

The hyperbolic cosine function and hyperbolic sine function have the followingproperties.

d

dxcoshx = sinhx, (161)

d

dxsinhx = coshx. (162)

28

Then, analogously, we find that {an[coshx]} and {an[sinhx]} have the followingproperties.

DR{an[coshx]} = {(22n+1 − 1)/2n+2} =1

4,7

8,31

16, · · · = {an[sinhx]}+ {1/(2n+2)},

(163)

DR{an[sinhx]} = {(22n+1 + 1)/2n+2} =3

4,9

8,33

16, · · · = {an[coshx]} − {1/(2n+2)}.

(164)

Also, the following relation holds for the hyperbolic functions.

cosh2 x− sinh2 x = 1. (165)

Analogously, the following relation holds for {an[coshx]} and {an[sinhx]}.

{an[coshx]}2 − {an[sinhx]}2 = {1}. (166)

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

cosh2 x+ sinh2 x = cosh 2x, (167)

2 coshx sinhx = sinh 2x. (168)

Analogously, the following relation holds for {an[coshx]} and {an[sinhx]}.Proposition 24 (Sequence version of addition theorem for hyperbolic function).

{an[coshx]}2 + {an[sinhx]}2 = {a2n[coshx]}, (169)

2{an[coshx]}{an[sinhx]} = {a2n[sinhx]}. (170)

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

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

{an[coshx]}nat =1

2[{an[[ex]]}+ {an[e−x]}nat] = 1, 1, 2, 4, 8, · · · , (171)

{an[sinhx]}nat =1

2[{an[[ex]]} − {an[e−x]}nat] = 0, 1, 2, 4, 8, · · · . (172)

As an analogy of Eqs. (161) and (162), {an[coshx]}nat and {an[sinhx]}nat havethe following properties.

DR{an[coshx]}nat = {an[sinhx]}nat, (173)

DR{an[sinhx]}nat = {an[coshx]}nat. (174)

Also, {an[cosx]}nat and {an[sinx]}nat have the following property analogous to Eq. (165).

({an[coshx]}nat)2 − ({an[sinhx]}nat)2 = {an[e−x]}nat. (175)

Furthermore, the {an[coshx]}nat and {an[sinhx]}nat satisfy the following analogy ofthe addition theorem (Eq. (167) and (168)).

29

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

({an[coshx]}nat)2 + ({an[sinhx]}nat)2 = {a2n[coshx]}nat, (176)

2{an[coshx]}nat{an[sinhx]}nat = {a2n[sinhx]}nat. (177)

Figure 12(c) shows difference between {an[coshx]} ({an[sinhx]}nat) and {an[coshx]}nat({an[sinhx]}).

30

7 Trigonometric function

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

eix = cosx+ i sinx, (178)

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

Definition 25 (Right sequence version of trigonometric function).

{an[cosx]}R = {Re(1 + i)n} = 1, 1, 0,−2,−4,−4, 0, 8, 16, 16, · · · , (179)

{an[sinx]}R = {Im(1 + i)n} = 0, 1, 2, 2, 0,−4,−8,−8, 0, 16, · · · . (180)

Figure 13(a) shows a graph of ({an[cosx]}R, {an[sinx]}R) that represents thedivergent spiral. This contrasts with the circle represented by (cos x, sinx).

-3

-2

-1

0

1

2

3

-3 -2 -1 0 1 2 3

(an [cos x]R, an [sin x]R )

(a)

Re

Im

0 1 32-3 -2 -1

0

-1

-2

-3

1

3

2

-1

-0.5

0

0.5

1

-1 -0.5 0 0.5 1

(an [cos x]L, an [sin x]L )

(b)

Re

Im

0 1 32-3 -2 -1

0

-1

-2

-3

1

3

2

Figure 13: Graph of (a) ({an[cosx]}R, {an[sinx]}R) corresponding to the divergentspiral and (b) ({an[cosx]}L, {an[sinx]}L) corresponding to the convergent spiral.

The cosine function and sine function have the following properties for differen-tial.

d

dxcosx = − sinx, (181)

d

dxsinx = cosx. (182)

Then, analogously, we find that {an[cosx]}R and {an[sinx]}R have the followingproperties.

DR{an[cosx]}R = 0,−1,−2,−2, 0, 4, 8, 8, 0,−16, · · · = −{an[sinx]}R, (183)

DR{an[sinx]}R = 1, 1, 0,−2,−4,−4, 0, 8, 16, 16, · · · = {an[cosx]}R. (184)

31

Also, the following relation holds for the trigonometric functions.

cos2 x+ sin2 x = 1. (185)

Analogously, the following formula for {an[cosx]}R and {an[sinx]}R holds.

{an[cosx]}2R + {an[sinx]}2R = 2n. (186)

In addition, the tangent function is given by

tanx =sinx

cosx. (187)

Then, the sequence version of the tangent function determined by {an[cosx]}R and{an[sinx]}R is given by

{an[tanx]}R ={an[sinx]}R{an[cosx]}R

= 0, 1,∞,−1, 0, 1,−∞,−1, 0, 1, · · · . (188)

where we wrote n/0 =∞ and −n/0 = −∞ for a natural number n.Furthermore, the trigonometric functions satisfy an addition theorem as follows.

cos 2x = cos2 x− sin2 x, (189)

sin 2x = 2 sin x cosx. (190)

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

Proposition 26 (Addition theorem for right sequence version of trigonometric func-tion).

{an[cosx]}2R − {an[sinx]}2R = 1, 0,−4, 0, 16, 0, · · · = {a2n[cosx]}R, (191)

2{an[cosx]}R{an[sinx]}R = 0, 2, 0,−8, 0, 32, · · · = {a2n[sinx]}R. (192)

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

Definition 26 (Left sequence version of trigonometric function).

{an[cosx]}L = {Re(1− i)−n} = 1,1

2, 0,−1

4,−1

4,−1

8, 0,

1

16

1

16,

1

32, · · · , (193)

{an[sinx]}L = {Im(1− i)−n} = 0,1

2,1

2,1

4, 0,−1

8,−1

8,− 1

16, 0,

1

32, · · · . (194)

Figure 13(b) shows a graph of ({an[cosx]}L, {an[sinx]}L) that represents theconvergent spiral. This contrasts with the circle represented by (cosx, sinx), andalso the divergent spiral represented by ({an[cosx]}R, {an[sinx]}R).

As an analogy of Eqs. (181) and (182), {an[cosx]}L and {an[sinx]}L have thefollowing properties.

DL{an[cosx]}L = 0,−1

2,−1

2,−1

4, 0,

1

8,1

8,

1

16, 0,− 1

32, · · · = −{an[sinx]}L, (195)

DL{an[sinx]}L = 1,1

2, 0,−1

4,−1

4,−1

8, 0,

1

16

1

16,

1

32, · · · = {an[cosx]}L, (196)

32

where we assumed a−1[cosx]L = 1 and a−1[sinx]L = −1.Also, {an[cosx]}L and {an[sinx]}L have the following property that is analogous

to Eq. (185).

{an[cosx]}2L + {an[sinx]}2L = 2−n. (197)

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

{an[tanx]}L ={an[sinx]}L{an[cosx]}L

= 0, 1,∞,−1, 0, 1,−∞,−1, 0, 1, · · · = {an[tanx]}R.(198)

Furthermore, {an[cosx]}L and {an[sinhx]}L satisfy the following formula that isanalogous to the addition theorem (Eq. (189) and (190)).

Proposition 27 (Addition theorem for left sequence version of trigonometric func-tion).

{an[cosx]}2L − {an[sinx]}2L = 1, 0,−1

4, 0,

1

16, 0, · · · = {a2n[cosx]}L, (199)

2{an[cosx]}L{an[sinx]}L = 0,1

2, 0,−1

8, 0,

1

32, · · · = {a2n[sinx]}L. (200)

The following equality is called Euler’s identity and connects the Napier’s con-stant (e), the additive identity (0), the multiplicative identity (1), the imaginaryunit (i), the circle ratio (π) , and would be one of the most famous and beautifulequality in mathematics.

eiπ + 1 = 0. (201)

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

eiπ + 12 = 0, (202)

eiπ + 1−2 = 0. (203)

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

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

a4[eix]R + 22 = 0, (204)

a4[eix]L + 2−2 = 0. (205)

Figure 14(a) shows schematic representation of Eqs. (204) and (205).Equation (204) is obtained by replacing eiπ by a4[e

ix]R and 12 by 22 in Eq. (202).Also, Eq. (205) is obtained by replacing eiπ by a4[e

ix]L and 1−2 by 2−2 in Eq. (203).In addition, as an analogy of the relation (see Fig. 3(c))

eiπeiπ = 1, (206)

33

Re

Im

0+2

+1/2

1/2

2

�p

�a4[eix]R

�p�a4[eix]L

�p

�a4[eix]R�p�a4[eix]L

Figure 14: (a) Sequence version of Euler’s identity (a4[eix]R + 22 = 0 and a4[e

ix]L +2−2 = 0), (b) a4[e

ix]La4[eix]R = 1.

the following formula holds.

a4[eix]La4[e

ix]R = 1. (207)

Figure 14(b) shows schematic representation of Eq. (207).Also, by combining {an[cosx]}R ({an[sinx]}L) and {an[cosx]}L ({an[sinx]}L)

generally, one has the following sequence version of cos x and sinx whose periodicityis 8.

{an[cosx]}per = {sgn (an[cosx]R)} × {an[cosx]}R × {an[cosx]}L (208)

= 1,1

2, 0,−1

2,−1,−1

2, 0,

1

2, · · · , (209)

{an[sinx]}per = {sgn (an[sinx]R)} × {an[sinx]}R × {an[sinx]}L (210)

= 0,1

2, 1,

1

2, 0,−1

2,−1,−1

2, 0, · · · . (211)

This periodicity is analogous to 2π-periodicity of cosx and sinx. In Fig. 15, we showrepresentation of (cosx, sinx) by the unit circle and of (an[cosx]per, an[sinx]per) bythe eight points on the square such that the length of the diagonal is 2. While the2π is the periodicity of cosx and sin x and the circumference of a unit circle, andthe 8 is the periodicity of {an[cosx]} and {an[sinx]} and the circumference lengthof the square when the interval of the two neighboring points is the unit of measure.

34

(b)

0 1

1

-1

-1

(an [cos x]per, an [sin x]per )

(a)

(cos x, sin x )

0 1

1

-1

-1

Figure 15: (a) Representation of (cosx, sinx) by the unit circle. (b) Representationof (an[cosx]per, an[sinx]per) by the eight points on the square such that the lengthof the diagonal is 2.

8 Fibonacci sequences

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

8.1 Fibonacci sequence

Fibonacci sequence is defined by

{Fn} = 0, 1, 1, 2, 3, 5, 8, · · · , (212)

and satisfies the relation

limn→∞

FnFn−1

= φ, (213)

where φ is the golden ratio and also the relation

Fn+2 = Fn+1 + Fn. (214)

This relation can be rewritten by using the right differential DR defined in Eq. (52)as follows.

DR{Fn+1} = {Fn}. (215)

Also, by using the shift operator Sk defined in Eq. (42), one has

DRS1{Fn} = {Fn}. (216)

This is similar to Eq. (91) that the right sequence version of exponential functionsatisfies. Then, we obtain the following theorem by using the insertion operator Iadefined in Eq. (50) that is similar to Eq. (4) (see Fig. 16 and compare Fig. 17 withFig. 8).

35

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

{Fn} = I0{an[1]}+ I20{an[x]}+ I30{an[x2/2]}+ · · · =∞∑

n=0

In+10 {an[xn/n!]}. (217)

1 1 1 1 1 1 ・・・

0 0 1 2 3 4 ・・・

・・・

0 0 0 0 1 3 ・・・

0 0 0 0 0 0 ・・・

+++

1 1 2 3 5 8

・・・

・・・

・・・

・・・

・・・

{an[1]}{an[x]}

・・・

I0

I20

I30

15

61

13 {Fn}

I40

00

・・・

0

0

0

・・・

{an[x2/2]}L

{an[x3/6]}L

Figure 16: Digrammatic representation of sequence version of Maclaurin series forFibonacci sequence corresponding to Pascal’s triangle.

Also, the negafibonacci sequence [25] is defined by

{F−n } = 0, 1,−1, 2,−3, 5,−8, · · · = {Fn} × {(−1)n+1}, (218)

and satisfies the following relation.

DR{F−n } = −S2{F−n }. (219)

Although this is similar to e−x, however the relation {Fn}×{F−n } = {an[1]} does nothold. Instead the following relation holds that is equivalent to Cassini’s Fibonacciidentity [26].

{Fn} × S2{F−n }+ S1{Fn} × S1{F−n } = {an[1]}. (220)

For example, F1F−3 + F−2 F2 = 1× 2 + 1× (−1) = 1.

8.2 (P, Q)-Fibonacci sequence

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

{Fn(P,Q)} = 0, 1, P, P 2 +Q,P 3 + 2PQ,P 4 + 3P 2Q+Q2, · · · , (221)

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

{Fn+2(P,Q)} = P{Fn+1(P,Q)}+Q{Fn(P,Q)}. (222)

We find the (P, Q)-Fibonacci sequence has the following sequence version of Maclau-rin series (Fig. 18).

36

{Fn} = 1, 1, 2,

{an [1]}

{an [x]}

{an [x2/2]}L

{an [x3/6]}L

3, 5, 8, 13, ・・・

Figure 17: Diagrammatic representation of sequence version of Maclaurin series forFibonacci sequence by using blocks.

Proposition 28 (Sequence version of Maclaurin series for (P,Q)-Fibonacci se-quence).

{Fn(P,Q)} = I0[{an[1]} × {P n}] +QI20 [{an[x]} × {P n}] + · · · , (223)

=∞∑

k=0

Ik+10 Qk[{an[xk/k!]} × {P n}]. (224)

1 P P2 P3 P4 ・・・

0 0 Q 2PQ 3P2Q ・・・

・・・

0 0 0 0 Q2 ・・・

++

1 P2 + Q

・・・

・・・

・・・

・・・

・・・

00

・・・

0

0 P P3 + 2PQ P4 +3P2Q +Q2

・・・

{Fn(P, Q)}

I30Q2[{an[x2/2]}L ⇥ {Pn}]

I0[{an[1]} ⇥ {Pn}]

I20Q[{an[x]} ⇥ {Pn}]

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 thefollowing relation.

limn→∞

Fn(2, 1)

Fn−1(2, 1)= 1 +

√2, (225)

37

where 1 +√

2 is the silver ratio. The sequence version of Maclaurin series for PellSequence is shown in Fig. 19.

1 2 4 8 16 32 ・・・

0 0 1 4 12 32 ・・・

・・・

0 0 0 0 1 6 ・・・

++

1 2 5 12 29 70

・・・

・・・

・・・

・・・

・・・

・・・

00

・・・

0

0

・・・

{Fn(2, 1)}

I30 [an[x2/2]L ⇥ {2n}]

I20 [{an[x]} ⇥ {2n}]

I0[{an[1]} ⇥ {2n}]

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 thesequence version of Maclaurin series for Jacobsthal Sequence is shown in Fig. 20.

1 1 1 1 1 1 ・・・

0 0 2 4 6 8 ・・・

・・・

0 0 0 0 4 12 ・・・

++

1 1 3 5 11 21

・・・

・・・

・・・

・・・

・・・

・・・

00

・・・

0

0

・・・

{Fn(1, 2)}

2I0{an[x]}L

4I20{an[x2/2]}L

I0{an[1]}L

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] sat-isfying the following relation.

{Fn+k(k)} =k−1∑

l=0

{Fn+l(k)}. (226)

Then, we construct the sequence version of Maclaurin series for {Fn+k(k)}. First,we define deformed integral by combining the integral I0 and the insertion I0.

38

Definition 27 (Deformed integral). For an integer k ≥ 2,

Ik =k−2∑

l=0

I l0I0L. (227)

Then, we have the following proposition.

Proposition 29 (Sequence version of Maclaurin series for k-bonacci sequence). Foran integer k such that k ≥ 2,

{Fn(k)} = Ik−10 {an[1]}+ IkIk0 {an[1]}+ I2kIk+10 {an[1]}+ · · · (228)

=∞∑

l=0

(Ik)lIk−1+l0 {an[1]}. (229)

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

Definition 28 (Limit of sequence). For an integer k and a sequence {an} dependingon k,

limk→∞{an(k)} = lim

k→∞a0(k), lim

k→∞a1(k), · · · . (230)

Then, we obtain the following relation that states Sk{Fn(k)} approaches to{an[ex]}R in the limit of k →∞.

Proposition 30 (Limit of k-bonacci sequence and right sequence version of expo-nential function).

limk→∞Sk{Fn(k)} = {an[ex]}R. (231)

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

I2I30{an[1]} = I0L[I30{an[1]}+ I40{an[1]}] = I0L(0, 0, 0, 1, 2, 2, 2, · · · )= 0, 0, 0, 0, 1, 3, 5, 7, · · · . (232)

39

0 1 1 1 1 1 ・・・

0 0 0 1 3 5 ・・・

・・・

0 0 0 0 0 1 ・・・

++

0 1 1 2 4 7

・・・

・・・

・・・

・・・

・・・

・・・

00

・・・

0

0

・・・

I20an[1]

{Fn(3)}

I2I30an[1]

I22I4

0an[1]

Figure 21: Digrammatic representation of sequence version of Maclaurin series fortribonacci sequence ({Fn(3)}).

9 Factorial and Bell number

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

9.1 Factorial

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

n! = n× (n− 1)× · · · × 2× 1, (233)

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

n! =

∫ ∞

0

xn−1e−xdx, (234)

we construct sequence dual of factorial as follows.

{an〈n!〉} =∞∑

k=0

kn × 2−k, (235)

= 2, 2, 6, 26, 150, 1082, 9366, 94586, 1091670, 14174522, · · · . (236)

We find that {an〈n!〉} deeply relates to the factorial n!. First, we focus on thefollowing transform for a sequence {an} called Stirling transform [37].

{bn} =

{n∑

k=0

S(n, k)ak

}, (237)

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

Then, the following proposition holds.

40

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

S1{an〈n!〉} = 2

{n∑

k=0

S(n, k)k!

}. (238)

This proposition shows the sequence dual of factorial is equivalent to the Stirlingtransform of the second kind for the factorial up to the shift operation and scalarmultiplication.

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

n! ≡ −1 (mod n+ 1), (239)

if and only if n + 1 is a prime. For example, 4 × 3 × 2 × 1 = 24 = 5 × 5 − 1,5× 4× 3× 2× 1 = 120 = 6× 20− 0. Then, we obtain the sequence dual of Wilson’stheorem as follows.

Theorem 4 (Sequence dual of Wilson’s theorem). For a natural number n,

{an〈n!〉} ≡ 0 (mod n+ 1), (240)

if n+ 1 is a prime number.

For example, {a6〈n!〉} = 9366 = 1338 × 7 and {a8〈n!〉} = 1091670 = 121297 ×9− 3. To prove the above theorem, it is enough to see the following formula holdsfor a prime number n and a natural number k that is less than or equal to n− 1,

k!× Sn−1,k ≡ −(−1)k mod n, (241)

because the above formula leads to∑n−1

k=1 k! × Sn−1,k ≡ 0 mod n. By using thefollowing formula,

k!× Sn−1,k =k∑

i=0

(−1)ikCi(k − i)n−1, (242)

and the Fermat’s little theorem that states ln−1 ≡ 1 mod n if l is not divisible by aprime number n, one can show Eq. (241). Figure 22 shows the example of Eq. (241)for n = 7.

2 6 24 12031 90 65 1+

62 540 1560 1800

11

1

k!720S(6, k)15

720

≡ 1 ≡ -1 mod 7≡ 1 ≡ -1 ≡ 1 ≡ -1

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

We remark that Eq. (240) holds for some odd composite numbers. For example,{a24〈n!〉} can be divided by 25. Also, for all the even numbers, Eq. (240) does nothold because

∑n−1k=2 k!× Sn−1,k mod n is an even number mod n and does not equal

to −1 mod n.

41

9.2 Bell number

Bell number is a number counting how many number of ways to partition of a set ofn 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 B3

is 5. The nth term of Bell numbers is given by

Bn =n−1∑

k=0

nCkBk. (243)

Then, one has

{Bn} = 1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147, · · · . (244)

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

Bn =n∑

k=0

Sn,k. (245)

Also, the unsigned Stirling number of the first kind |s(n, k)| is a number that countsthe 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 theone element: (1,2) or (2, 3) or (3, 1). Then, the factorial can be represented asfollows,

n! =n∑

k=0

|sn,k|. (246)

Now, we construct the sequence dual of Bell numbers. First, the exponentialgenerating function of the nth Bell number is given by

∞∑

n=0

Bn

n!xn = ee

x−1. (247)

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

∞∑

n=0

Bn+1

n!xn = exee

x−1. (248)

Note that ex can be obtained by substituting α = 1 into eαx. Then, we define thesequence dual of Bell numbers by using a1[e

αx]L that can be obtained by substitutingn = 1 into {an[eαx]}L.

∞∑

n=0

an〈Bn〉n!

αn =1

1− αe( 11−α−1) = a1[e

αx]L e(a1[eαx]L−1). (249)

We remark that {an〈Bn〉} relates to Bn through the Stirling transform of the firstkind corresponding to Eq. (238) [37].

42

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

{an〈Bn〉} =

{n∑

k=0

|s(n, k)|S1Bk

}, (250)

= 1, 2, 7, 34, 209, 1546, 13327, 130922, 1441729, 17572114, · · · . (251)

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

Bn+p −Bn ≡ Bn+1 (mod p). (252)

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

Theorem 5 (Sequence dual of modular property of Bell numbers). For naturalnumbers n and m,

{an+m〈Bn〉} − {an〈Bn〉} ≡ 0 (mod m). (253)

For example, a7〈Bn〉−a6〈Bn〉 = 130922−13327 = 1×117595, a7〈Bn〉−a5〈Bn〉 =130922− 1546 = 2× 64688, a7〈Bn〉− a4〈Bn〉 = 130922− 209 = 3× 43571, a7〈Bn〉−a3〈Bn〉 = 130922 − 34 = 4 × 32722, a7〈Bn〉 − a2〈Bn〉 = 130922 − 7 = 5 × 26183,a7〈Bn〉−a1〈Bn〉 = 130922−2 = 6×21820, a7〈Bn〉−a0〈Bn〉 = 130922−1 = 7×18703.

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

an〈Bn〉 = 2nan−1〈Bn〉 − (n− 1)2an−2〈Bn〉, (254)

and induction method as follows.

an+1+m〈Bn〉 ≡ 2(n+ 1)an〈Bn〉 − n2an−1〈Bn〉 mod m ≡ an+1〈Bn〉 mod m. (255)

Note that for the first two terms of {an〈Bn〉}, the following relations hold.

am〈Bn〉 ≡ a0〈Bn〉 mod m ≡ 1 mod m, (256)

am+1〈Bn〉 − a1〈Bn〉 ≡ a1〈Bn〉 mod m ≡ 2 mod m. (257)

The first relation can be proved by the following formula for {an〈Bn〉}.

an〈Bn〉 =n∑

k=0

k!(nCk)2. (258)

The second formula can be shown as follows.

a1+m〈Bn〉 = 2(1 +m)am〈Bn〉 − (1 +m− 1)2am−1〈Bn〉 ≡ 2(1 +m) mod m

≡ 2 mod m. (259)

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10 Discussion

In this paper, we created a sequence version of calculus. First, we defined equiva-lence, some fundamental operations, left/right differential and integral for sequences.Then, we proposed sequence version of power function, exponential function, hy-perbolic function, and trigonometric function and sequence versions of Maclaurinseries for them. We found the sequence versions of Maclaurin series for exponentialfunction involve divergent series including Grandi’s series. We remark that gener-ally there are some candidates of the sequence version of a given function. As forsequence versions of trigonometric functions, first, two kinds of sequence versionsof trigonometric functions were proposed, depending on whether the left or rightdifferential was used. The right/left trigonometric functions are graphically repre-sented by the divergent/convergent spiral while the usual trigonometric functionsare graphically represented by a circle. Then, we found the sequence version ofbinomial theorem and Euler’s identity. The sequence version of Euler’s identity iscomposed of two equations determined by fourth term of left/right sequence versionof eix. Then, by combining the two sequence versions of trigonometric functions, weproposed another sequence version of trigonometric functions whose periodicity is8. Furthermore, we proposed sequence version of Maclaurin series for Fibonacci se-quence, (P, Q)-Fibonacci sequence, and k-bonaaci sequence, and found similaritiesbetween them and the sequence versions of exponential function. In addition, weintroduce sequence dual of factorial and Bell numbers. Then, the sequence dual ofthe modular property of factorial concerning the prime numbers (Wilson’s theorem)and of the Bell numbers concerning the prime numbers were obtained. Finally, weremark a few points for future research. First, probably, our theory relates system-atically to the figurate number including (centered) polygonal numbers, (centered)pyramidal numbers, or (centered) Platonic numbers. Note that {an[x2/2]}R is thetriangle number. Also, there are a few functions that we could not deal with inthis paper such as logarithmic function, inverse trigonometric/hyperbolic function,elliptic integral, zeta function. Sequence version of these functions may lead to newformulae for some sequences. Also, considering other frameworks of analogy betweensequence and analytics may lead to many new theorems of sequences and analytics.

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A List of sequence version of functions and se-

quence duals

Constant function:{an[a]} = a, a, a, a, a, a, a, a, a, a, a, a, · · · .Power function:{an[xk]} = 0k, 1k, 2k, 3k, 4k, 5k, 6k, 7k, 8k, 9k, 10k, 11k, · · · .Exponential function (right):{an[eαx]}R = (1 + α)0, (1 + α)1, (1 + α)2, (1 + α)3, (1 + α)4, (1 + α)5, (1 + α)6,(1 + α)7, (1 + α)8, (1 + α)9, (1 + α)10, (1 + α)11, · · · .Exponential function (left):{an[eαx]}L = (1−α)0, (1−α)−1, (1−α)−2, (1−α)−3, (1−α)−4, (1−α)−5, (1−α)−6,(1− α)−7, (1− α)−8, (1− α)−9, (1− α)−10, (1− α)−11, · · · .Exponential function (natural):{an[e−x]}nat = 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, · · · .Hyperbolic cosine function:{an[coshx]} = 1, 5

4, 17

8, 6516, 257

32, 1025

64, 4097

128, 16385

256, 65537

512, 262145

1024, 1048577

2048, 4194305

4096, · · · .

Hyperbolic sine function:{an[sinhx]} = 1, 3

4, 15

8, 6316, 255

32, 1023

64, 4095

128, 16383

256, 65535

512, 262143

1024, 1048575

2048, 4194303

4096, · · · .

Hyperbolic cosine function (natural):{an[coshx]}nat = 1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, · · · .Hyperbolic sine function (natural):{an[sinhx]}nat = 0, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, · · · .Cosine function (right):{an[cosx]}R = 1, 1, 0,−2,−4,−4, 0, 8, 16, 16, 0,−32, · · · .Sine function (right):{an[sinx]}R = 0, 1, 2, 2, 0,−4,−8,−8, 0, 16, 32, 32, · · · .Tangent function (right):{an[tanx]}R = 0, 1,∞,−1, 0, 1,−∞,−1, 0, 1,∞,−1, · · · .Cosine function (left):{an[cosx]}L = 1, 1

2, 0,−1

4,−1

4,−1

8, 0, 1

16, 116, 132, 0,− 1

64· · · .

Sine function (left):{an[sinx]}L = 0, 1

2, 12, 14, 0,−1

8,−1

8,− 1

16, 0, 1

32, 132, 164, · · · .

Tangent function (left):{an[tanx]}L = 0, 1,∞,−1, 0, 1,−∞,−1, 0, 1,∞,−1, · · · .Cosine function (periodic){an[cosx]}per = 1, 1

2, 0,−1

2,−1,−1

2, 01

2, 1, 1

2, 0,−1

2, · · · .

Sine function (periodic){an[sinx]}per = 0, 1

2, 1, 1

2, 0,−1

2,−1,−1

2, 0, 0, 1

2, 1, 1

2, · · · .

Factorial:{an〈n!〉} = 2, 2, 6, 26, 150, 1082, 9366, 94586, 1091670, 14174522, 204495126,3245265146, · · · .Bell number:{an〈Bn〉} = 1, 2, 7, 34, 209, 1546, 13327, 130922, 1441729, 17572114, 234662231,3405357682, · · · .

45

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