Discrete Calculus and its Applications

Transcription

1 Discrete Calculus and its Applications Alexander Payne Princeton University Abstract We introduce the foundations of discrete calculus, also known as the calculus of finite differences, as an elegant, alternative approach to the computation of sums. Then, we will derive several discrete analogs of basic theorems from infinite calculus, such as the product rule and summation by parts, to help in computing explicit formulae for nontrivial sums. Finally, we conclude with an application 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. We want to find an expression for this in terms of. Consider ( + 1). ( + 1) times (0 + ) + (1 + ( )) + + (( 2) + 1) + (( ) + 1) + ( + 0) Now, we can see that the first terms in each of the parentheses consists of exactly the terms in, and the same goes for the second term in each of the parentheses. Thus, From this, we conclude that ( + 1) + 2 ( + 1) 2 That took a bit of work for such a simple sum! It would certainly be much more difficult to find 2, let alone 2. A systematic approach to finding expressions for finite (and infinite) sums is clearly desirable, particularly one that will not depend so heavily on exactly what we re summing. It turns out that discrete calculus is exactly what we want and more. 33

2 2 Foundations of Discrete Calculus In this section and for the rest of these notes, we will work with sequences { }, where each term of the sequence is a real number. In analogy with traditional calculus, we define the discrete derivative of this sequence, generally following the ideas in [1]. Definition 2.1. For a sequence { }, the forward difference (also called a discrete derivative) of is a sequence, which is defined by { } The -th term of is denoted by, which is defined as +1. For a specific example, let { }. Then, ( 2. The sequence given by {2 } is unique (up to a constant) in that it is a nontrivial sequence that is fixed under application of the forward difference. The forward difference measures how a sequence changes since if we consider the points (0 0 ) (1 1 )( ) in the Cartesian plane, ( + 1) So, is the slope of the line connecting the points ( ) and ( ). The forward difference has 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 { }, 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 { } and { } where, Also, Proof lim ( ) ( )+ +( Now, using the above to get the infinitary case of the Fundamental Theorem of Discrete Calculus, lim lim ( ) 1 34

3 These theorems mean that we have just transformed the problem of computing to the problem of finding a sequence such that. For example, we may now compute for any real constant 1using this theorem. We find that () +1 ( 1), so using Lemma 3.2 and dividing over to the other side (the 1condition is used here), we get ( ) Thus, using the Fudamental Theorem of Discrete Calculus, we find that for 1and any R, Then, for < 1, lim lim lim Now, we move on to theorems which will greatly expand the types of sums which we can compute using these methods. Theorem 2.4 (Discrete Calculus Product Rule). Let { } and { } be sequences, and define { }. Then, ()( ) +1 + ( ). Proof. () ( ) +1 + ( ) Now we may sum the product rule to get a very nice expression. Theorem 2. (Summation by Parts). Let { } and { } be sequences. Then, ( )[ ] ( ) +1 Proof. By summing the product rule and applying the Fundamental Theorem of Discrete Calculus, we get and ()[ ] () [( ) +1 + ( )] [( ) +1 ]+ ( ) Then, by solving for ( ), we get ( )[ ] ( ) +1 3

4 This allows us to compute more complicated sums such as. Let and with 1, and apply the summation by parts formula to get + 1)+1 ( )[( 0] (( + 1) ) +1 ( + 1) +1 ( + 1)+1 +1 ( + 1)+1 (+1 ) (+2 ) 1 (( + 1) ) In particular, 2 ( ) Finally, to compute a sum such as 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 Falling factorials are nice because ( )( 2) ( + 2)( + 1) ( + 1) ( + 1)( + 2) ( ) ( + 1) ( +1 + )( ) ( + 1) 1 Note that falling factorials follow a power rule similar to that of monomials in infinite calculus, so they allow us to compute many more types of sums. For example, we may write , which allows us to easily compute the sum of 2. 2 ( )( 1 3 ( + 1) )+( 1 2 ( + 1)2 2 0 ( + 1)(2 + 1) 6 3 Discrete Calculus and the Fibonacci Numbers Now, we present an application of the above material to the computation of an explicit formula for the Fibonacci numbers, which is a clarification of this derivation using discrete calculus found in [2]. We define a sequence {F 0 F 1 F 2 F 3 F 4 }, where F 0 1and F 1 1and successive terms are constructed using the recurrence relation F +2 F +1 + F. This sequence is known as the Fibonacci sequence, and its first few terms are { }. We will find an expression for the -th term of 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 F +2 2F +1 + F +1 + F 2F (F +2 2F +1 + F )+(F +1 F ) F ( F + F F 36

5 The operator 2 returns the sequence obtained from applying the forward difference twice. This means that any sequence satisfying the recurrence relation must also satisfy the expression with the forward 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 {F 0 F 1 F 2 F 3 F 4 } to {F 1 F 0 F 0 F 2 F 1 F 1 F 3 F 2 F 2 }. So, the -th term of ( ) is given by ( ) F F F. Consier the equation 2 +, which is the same form as the derived equation that we found above with the forward difference operators. The solutions to 2 + are We can then see that we may factor ( F + F F as ( ( 2 )( ( ))F 2 This just says that we apply the two operators successively to F and get a sequence with all terms zero. To check this, we compute ( ( 2 + )( ( 1 2 ) F ( ( 2 + ) [F +1 F ( 2 F ] The term in the brackets is just the -th term of a sequence to which we apply the operation ( ( 2 + ). (F +2 F +1 ( 2 F +1) (F +1 F ( 2 ( 2 + (F +1 F ( 2 F +2 F +1 F F ) F ) Thus, this factorization is legitimate. One solution to the factored formula we found would be if This is because ( ( 2 ) F ( ( 2 + )(0) 0 Now, we solve the first part of the factored formula. So, 0( ( 2 ) F F +1 F ( 2 F +1 ( 1 2 F Then, by iterating, we get a sequence that satisfies the Fibonacci relation. F ( 1 2 F 0 ( F

6 Similarly, we may write ( ( 2 + )( ( 1 2 ))F 2 And we get another sequence that satisfies the Fibonacci relation. F ( Then, since constants can be removed from inside a foward difference and the forward difference of a sum 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( D( 1 2 We know the starting conditions F 0 F 1 1, so we get and Solving for C and D, we find 1F 0 C( D( C + D 1F 1 C( D(1 2 C D So, by plugging these back in, we get the expression for the -th term of the Fibonacci sequence. 4 Concluding Remarks F ( )( ( )(1 2 1 ( ( ( ( ( 1 2 ( Discrete calculus and standard infinite calculus have multiple parallels that allow for techniques to be translated between these two domains. Discrete calculus has a wide range of applications to disparate branches of math, such as discrete dynamical systems and complex analysis. Even applications to number theory have been found. Gilbreath s conjecture is a very famous open problem in number theory that can be 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 1the 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 inspire the reader to continue to study this subject and apply it to new branches of math. 38

7 References [1] David Gleich. Finite Calculus: A Tutorial for Solving Nasty Sums. (200). [2] George Kunin. The Finite Difference Calculus and Applications to the Interpolation of Sequences. MIT Undergraduate Journal of Mathematics [3] Walter Rudin. Principles of Mathematical Analysis. pg. 78. Theorem