Discrete Calculus and its Applications

Alexander PaynePrinceton UniversityAbstractWe introduce the foundations of discrete calculus, also known as the calculus of finite differ-ences, as an elegant, alternative approach to the computation of sums. Then, we will derive severaldiscrete analogs of basic theorems from infinite calculus, such as the product rule and summationby parts, to help in computing explicit formulae for nontrivial sums. Finally, we conclude with anapplication of discrete calculus to the Fibonacci sequence.

1 Motivation for Discrete Calculus

Consider the most basic of sums, the sum of all integers from 0 to � .��

�=0 �

We want to find an expression for this in terms of � . Consider � (� + 1).� (� + 1) = � + � + · · · + �� �� �

�+1 times= (0 + � ) + (1 + (� − 1)) + · · · + ((� − 2) + 1) + ((� − 1) + 1) + (� + 0)Now, we can see that the first terms in each of the parentheses consists of exactly the terms in��

�=0 �, and the same goes for the second term in each of the parentheses. Thus,� (� + 1) = ��

�=0 � + ���=0 � = 2 ��

�=0 �

From this, we conclude that��

�=0 � = � (� + 1)2That took a bit of work for such a simple sum! It would certainly be much more difficult to find�

�

�=0 �

2, let alone ��

�=0 �2�. A systematic approach to finding expressions for finite (and infinite) sumsis clearly desirable, particularly one that will not depend so heavily on exactly what we’re summing. Itturns out that discrete calculus is exactly what we want and more.

33

Principia: The Princeton Undergraduate Mathematics Journal2 Foundations of Discrete Calculus

In this section and for the rest of these notes, we will work with sequences � = {�0� �1� �2� ���}, whereeach term of the sequence is a real number. In analogy with traditional calculus, we define the discretederivative of this sequence, generally following the ideas in [1].Definition 2.1. For a sequence � = {�0� �1� �2� ���}, the forward difference (also called a discrete derivative)

of � is a sequence �� , which is defined by

�� = {�1 − �0� �2 − �1� �3 − �2� ���}

The �-th term of �� is denoted by �

�

� , which is defined as �

�

� = �

�+1 − �

�

.

For a specific example, let � = {2� 4� 8� 16� � � � � 2�

� � � � }. Then, �

�

� = 2�+1− 2� = 2�(2 − 1) = 2�.The sequence given by {2�

} is unique (up to a constant) in that it is a nontrivial sequence that is fixedunder application of the forward difference. The forward difference measures how a sequence changessince if we consider the points (0� �0)� (1� �1)� � � � � (�� �

�

)� � � � in the Cartesian plane,�

�

� = �

�+1 − �

�

= �

�+1 − �

�(� + 1) − �So, �

�

� is the slope of the line connecting the points (�� �

�

) and (� + 1� �

�+1). The forward differencehas a couple important linearity properties. This is the content of the following theorem.Lemma 2.2. For a sequence � and a sequence �, �(� + �) = �� + ��. Also, for a constant real number

�, if the sequence �� is defined by �� = {��1� ��2� ��3� ���}, then �(�� ) = � � � .

Proof. This proof is obvious and is left to the reader.It turns out that the forward difference is very useful in computing sums.

Theorem 2.3 (The Fundamental Theorem of Discrete Calculus). For any sequences � = {�0� �1� �2� ���}

and � = {�0� �1� �2� ���} where �� = � ,

���=0 �

�

= �

�+1 − �1Also,

∞��=0 �

�

= lim�→∞

(��

) − �1Proof.

���=0 �

�

= ���=0 �

�

� = �

�+1 + (��

− �

�

) + · · · + (�2 − �2) − �1 = �

�+1 − �1Now, using the above to get the infinitary case of the Fundamental Theorem of Discrete Calculus,

∞��=0 �

�

= lim�→∞

���=0 �

�

= lim�→∞

(��

) − �1

34

Principia: The Princeton Undergraduate Mathematics JournalThese theorems mean that we have just transformed the problem of computing �

�

�=0 �

�

to the problemof finding a sequence � such that �� = � . For example, we may now compute ��

�=0 �

� for any realconstant � �= 1 using this theorem. We find that ��

� = (�)�+1− �

� = (� − 1)��, so using Lemma 3.2 anddividing over � − 1 to the other side (the � �= 1 condition is used here), we get�( �

�

� − 1) = �

�

Thus, using the Fudamental Theorem of Discrete Calculus, we find that for � �= 1 and any � ∈ R,��

�=0 ��

� = ��

�+1� − 1 −

��

0� − 1 = �

�

�+1− 1

� − 1Then, for |�| < 1,

∞��=0 ��

� = lim�→+∞

���=0 ��

� = lim�→+∞

�

�

�+1− 1

� − 1 = lim�→+∞

�

�

�+1 + 11 − |�|

= �1 − �

Now, we move on to theorems which will greatly expand the types of sums which we can computeusing these methods.Theorem 2.4 (Discrete Calculus Product Rule). Let � = {�0� �1� �2� ���} and � = {�0� �1� �2� ���} be

sequences, and define �� = {�0�0� �1�1� �2�2� ���}. Then, �

�

(��) = (��

� )��+1 + �

�

(��

�).Proof.

�

�

(��) = �

�+1�

�+1 − �

�

�

�

= �

�+1�

�+1 − �

�

�

�+1 + �

�

�

�+1 − �

�

�

�= (��

� )��+1 + �

�

(��

�)Now we may sum the product rule to get a very nice expression.

Theorem 2.5 (Summation by Parts). Let � = {�0� �1� �2� ���} and � = {�0� �1� �2� ���} be sequences. Then,

���=0 �

�

(��

�) = [��+1�

�+1 − �0�0] − ���=0 (�

�

� )��+1

Proof. By summing the product rule and applying the Fundamental Theorem of Discrete Calculus, we get��

�=0 �

�

(��) = [��+1�

�+1 − �0�0]and

���=0 �

�

(��) = ���=0 [(�

�

� )��+1 + �

�

(��

�)] = ���=0 [(�

�

� )��+1] + ��

�=0 �

�

(��

�)Then, by solving for �

�

�=0 �

�

(��

�), we get��

�=0 �

�

(��

�) = [��+1�

�+1 − �0�0] − ���=0 (�

�

� )��+1

35

Principia: The Princeton Undergraduate Mathematics JournalThis allows us to compute more complicated sums such as �

�

�=0 ��

�. Let �

�

= � and ��

�

= �

� with� �= 1, and apply the summation by parts formula to get

���=0 ��

� = ���=0 � �

�

( �

�

� − 1) = [ (� + 1)��+1� − 1 − 0] − ��

�=0 ((� + 1) − �) �

�+1� − 1

= (� + 1)��+1� − 1 −

1� − 1 ��

�=0 �

�+1 = (� + 1)��+1� − 1 −

�

� − 1(��+1− 1

� − 1 )(� + 1)��+1

� − 1 −

1� − 1(��+2

− 1� − 1 − 1) = 1

� − 1((� + 1)��+1−

�

�+2− 1

� − 1 + 1)In particular, �

�

�=0 �2� = (� − 1)2�+1 + 2.Finally, to compute a sum such as ��

�=0 �

2, we must have one more definition.Definition 2.6. The �-th falling power (a.k.a. falling factorial) of n is denoted by �

�

, which is defined as

�

� = �(� − 1)(� − 2) � � � (� − � + 2)(� − � + 1)Falling factorials are nice because

��

� = (� + 1)� − �

� = (� + 1)� � � � (� − � + 2) − �(� − 1) � � � (� − � + 1)= (� + 1 − � + � − 1)�(� − 1) � � � (� − � + 1) = ��

�−1Note that falling factorials follow a power rule similar to that of monomials in infinite calculus, sothey allow us to compute many more types of sums. For example, we may write �

2 = �

2 + �

1, whichallows us to easily compute the sum of �

2.��

�=0 �

2 = ���=0 (�2 + �

1) = (13(� + 1)3 −

1303) + (12(� + 1)2 −

1202) = � (� + 1)(2� + 1)63 Discrete Calculus and the Fibonacci Numbers

Now, we present an application of the above material to the computation of an explicit formula forthe Fibonacci numbers, which is a clarification of this derivation using discrete calculus found in [2].We define a sequence {F0� F1� F2� F3� F4� � � � }, where F0 = 1 and F1 = 1 and successive terms areconstructed using the recurrence relation F

�+2 = F

�+1 + F

�

. This sequence is known as the Fibonaccisequence, and its first few terms are {1� 1� 2� 3� 5� 8� 13� � � � }. We will find an expression for the �-th termof the Fibonacci sequence using the principles of discrete calculus.Now, we may write the recurrence relation in terms of forward differences.F

�+2 = F

�+1 + F

�

F

�+2 − F

�+1 − F

�

= 0F

�+2 − 2F

�+1 + F

�+1 + F

�

− 2F

�

= 0(F�+2 − 2F

�+1 + F

�

) + (F�+1 − F

�

) − F

�

= 0(�2)�

F + �

�

F − F

�

= 036

Principia: The Princeton Undergraduate Mathematics JournalThe operator �

2 returns the sequence obtained from applying the forward difference twice. Thismeans that any sequence satisfying the recurrence relation must also satisfy the expression with theforward difference we derived. Now, we may factor this expression in a certain sense. For a constant c,(� − �) will be thought of as the operation that sends a sequence {F0� F1� F2� F3� F4� � � � } to {F1 − F0 −

�F0� F2 −F1 −�F1� F3 −F2 −�F2� � � � }. So, the �-th term of (�−�) is given by (�−�)�

F = �

�

F −�F

�

.Consier the equation �

2 + � − 1 = 0, which is the same form as the derived equation that we found abovewith the forward difference operators. The solutions to �

2 + � − 1 = 0 are� = −

12 −

√52 � −

12 + √52We can then see that we may factor (�2)�

F + �

�

F − F

�

= 0 as(� − (−12 −

√52 ))(� − (−12 + √52 ))F = 0This just says that we apply the two operators successively to F and get a sequence with all termszero. To check this, we compute

(� − (−12 + √52 ))(� − (−12 −

√52 ))�

F

= (� − (−12 + √52 ))�

[F�+1 − F

�

− (−12 −

√52 )F�

]The term in the brackets is just the �-th term of a sequence to which we apply the operation (� −(−12 + √52 )).= (F

�+2 − F

�+1 − (−12 −

√52 )F�+1) − (F

�+1 − F

�

− (−12 −

√52 )F�

)−(−12 + √52 )(F

�+1 − F

�

− (−12 −

√52 )F�

)= F

�+2 − F

�+1 − F

�Thus, this factorization is legitimate. One solution to the factored formula we found would be if(� − (−12 −

√52 ))�

F = 0This is because (� − (−12 + √52 ))(0) = 0Now, we solve the first part of the factored formula.

0 = (� − (−12 −

√52 ))�

F = F

�+1 − F

�

− (−12 −

√52 )F�

So,F

�+1 = (12 −

√52 )F�Then, by iterating, we get a sequence that satisfies the Fibonacci relation.

F

�

= (12 −

√52 )�F0 = (12 −

√52 )�37

Principia: The Princeton Undergraduate Mathematics JournalSimilarly, we may write (� − (−12 + √52 ))(� − (−12 −

√52 ))F = 0And we get another sequence that satisfies the Fibonacci relation.F

�

= (12 + √52 )�Then, since constants can be removed from inside a foward difference and the forward difference of asum is the sum of forward differences (Theorem 3.2), we get a general solution for the Fibonacci relation,where C and D are real constants.

F

�

= C (12 + √52 )� + D(12 −

√52 )�We know the starting conditions F0� F1 = 1, so we get

1 = F0 = C (12 + √52 )0 + D(12 −

√52 )0 = C + D

and 1 = F1 = C (12 + √52 ) + D(12 −

√52 )Solving for C and D, we findC = 12 + 12√5 � D = 12 −

12√5So, by plugging these back in, we get the expression for the �-th term of the Fibonacci sequence.F

�

= (12 + 12√5)(12 + √52 )� + (12 −

12√5)(12 −

√52 )�= 1

√5(12 + √52 )(12 + √52 )� + 1√5(12 −

√52 )(12 −

√52 )�= 1

√5(12 + √52 )�+1 + 1√5(12 −

√52 )�+1

4 Concluding Remarks

Discrete calculus and standard infinite calculus have multiple parallels that allow for techniques tobe translated between these two domains. Discrete calculus has a wide range of applications to disparatebranches of math, such as discrete dynamical systems and complex analysis. Even applications to numbertheory have been found. Gilbreath’s conjecture is a very famous open problem in number theory that canbe formulated in terms of discrete calculus. Using the notation of this paper,Conjecture 1 (Gilbreath’s Conjecture). Let � = {�

�

}, where �

�

is the n-th prime number. Then, define

for � = 1 the sequence {�

1�

} by �

1�

= �

�

�. For � > 1, define the sequence {�

�

�

} by �

�

�

= | �

�

�

�−1�

|.

Informally, Gilbreath’s conjecture says that the first term of the unsigned �-th forward difference of the

ordered sequence of prime numbers is 1 for all positive � . Then, for each � > 0, �

�1 = 1.Although this paper is far from complete in its survey of discrete calculus, the author hopes to inspirethe reader to continue to study this subject and apply it to new branches of math.38

Principia: The Princeton Undergraduate Mathematics JournalReferences

[1] David Gleich. Finite Calculus: A Tutorial for Solving Nasty Sums.https://www.cs.purdue.edu/homes/dgleich/publications/Gleich%202005%20-%20finite%20calculus.pdf(2005).[2] George Kunin. The Finite Difference Calculus and Applications to the Interpolation of Sequences. MITUndergraduate Journal of Mathematics. 101-109.[3] Walter Rudin. Principles of Mathematical Analysis. pg. 78. Theorem 3.55.

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