Abstract
We introduce a context-free grammar to generate Fibonacci and Lucas sequences. By applying the grammar , we give a grammatical proof of the Binet formula. Besides, we use the grammar to provide a unified approach to prove several binomial convolutions about Fibonacci and Lucas numbers, which were given by Hoggatt, Carlitz, and Church. Meanwhile, we also obtain some new binomial convolutions.
1. Introduction
Recall that the Fibonacci sequence and the Lucas sequence are defined through the same recurrence relations: for ,with initial values and , respectively.
Fibonacci and Lucas numbers have close connections with the golden ratio. Through this paper, we use to denote the golden ratio, that is, . Let . and are two roots of the quadratic equation . The famous Binet formulas for Fibonacci and Lucas numbers show that, for ,
The binomial identities involving Fibonacci and Lucas numbers are studied widely in recent decades. The study of binomial identities involving Fibonacci and Lucas numbers is beginning from a group of identities of Hoggatt [1]. After then, Carlitz [2] and Church and Bicknell [3] enriched the binomial identities family. The details of these identities will be expanded in the next section.
In this paper, we introduce a context-free grammar to describe Fibonacci and Lucas numbers. Let
We find that, for ,
Here, is the formal derivative associated with . By applying this grammar, we give a grammatical framework to prove the binomial convolutions by Hoggatt, Carlitz, Church, and Bicknell and obtain some new binomial convolutions involving Fibonacci and Lucas numbers.
In Section 2, we posed the binomial identities given by Hoggatt, Layman, Carlitz, Church, and Bicknell. In Section 3, we give a new context-free grammar, which is called Fibonacci grammar. Based on the grammar, we give a grammatical expression of Fibonacci and Lucas numbers. As the application of this expression, a grammatical proof of some classic relations associated with Fibonacci and Lucas numbers are given, including the Binet formula. In Section 4, we provide a uniform framework to prove binomial identities by a grammatical manner. We prove all the identities given by Hoggatt, Carlitz, Church, and Bicknell and give some new binomial identities associated with Fibonacci and Lucas numbers.
2. Binomial Identities of Hoggatt, Carlitz, Church, and Bicknell
In this section, we recall the work of Hoggatt, Carlitz, Church, and Bicknell on binomial identities about Fibonacci and Lucas numbers.
In [1], Hoggatt found that, for ,
Carlitz [2] extent Hoggatt’s series of identities to more general relations: for and ,
Besides, Church and Bicknell provided several different binomial convolutions about Fibonacci and Lucas numbers. In [3], Church and Bicknell showed that, for ,
3. Tilings and the Fibonacci Grammar
The approach of studying combinatorial polynomials by using context-free grammars was introduced by Chen [4]. In this decade, many combinatorists have found the relations between combinatorial polynomials and context-free grammars; see [4–7], for example. A context-free grammar is a set of substitution rules on a set of variables . We can define a formal derivative associated with a context-free grammar as a differential operator on polynomials or Laurent polynomials in . In precise, is a linear operator satisfying the relationwhich can be in general given as Leibnitz formula
For the purpose of combinatorial enumeration, the variables are attached to combinatorial structures by grammatical labelings by Chen and Fu [5]. In order to provide a combinatorial expression of Fibonacci numbers, we need a corresponding combinatorial structure. Although many combinatorial interpretations of Fibonacci numbers exist (see exercises 1–9 in [8], p.14, for example), we use the tiling definition given as below.
For , a tiling of length is refer to a tiling of a rectangle with size by squares and rectangles with size . Here, we call the rectangle with size by an -board, and call the rectangle with size by a domino.
For example, there are 5 tilings of length 4 Figure 1:
A classical result shows that there are exactly tilings with length for all . Here, we let count the unique empty tiling of 0-board. By counting the number of blocks of a tiling, we give a generating function as a -analogue of Fibonacci numbers:where denotes the number of tilings of an -board with blocks. Here, we define and .
Lemma 1. For , it holds that
Proof. Consider the last block of a tiling of length . There are two classes: ending with a square and ending with a domino. The first class corresponds to the function since the left tiling has size ; and the second class corresponds to the function . This completes the proof.
Letwhere is a constant, that is, . Since the grammar has close connections between the function , we call the grammar as the -Fibonacci grammar. Let in , the grammar degenerates to bewhich is called Fibonacci grammar.
Theorem 2. Let be the -Fibonacci grammar, and be the formal derivative associated with the grammar . For , we haveand for ,
Proof. It can be seen that (15) can be obtained from (14) by setting . To prove (14), we introduce a grammatical labeling of a tiling by labeling the blocks. We label each block of a tiling by , and label the last block extra by if it is a square, and by if it is a domino. Then, we define the weight of the tiling to be the sum of labelings of all blocks in , that is, for a tiling having blocks, when ending at a square and when ending at a domino. Then it is natural to say thatsince the sum of the weights of tilings ending at a square equals and the sum of the weights of tilings ending at a domino equals .
Now, we show by induction that, for , it holdswhere runs over the set of tilings with length .
For , , which is equal to the weight of the unique tiling of length 1. Thus, (17) holds for . Assume that (17) holds for . To show that (17) is valid for , we consider the process to generate a tiling of length from a tiling of length .
For a tiling ending at a domino, the only way to add the length of is adding a new square in the end of . In order to label the new tiling consistently, we delete the labeling for the last domino in old tiling and label the new square by . This corresponds to the substitution rule .
For a tiling ending at a square, we have two chooses. If we change the last square to a new domino, we change the labeling of the last part from to . If we add a new square at the end of , we turn the labeling from the old last square to the new last square and add a new labeling to it. These two chooses correspond to the substitution rule .
For example, the first tiling in Figure 1 is labeled by , and the unique corresponding tiling is labeled by .
Figure 2 And the second tiling in Figure 1 is labeled by , the two corresponding tilings are labeled by and , respectively Figure 3.
Notice that we can generate all tilings of length as above. Thus,where runs over the set of tilings with length . Thus (17) holds for . Now (17) holds for all by induction. This completes the proof.
(a)
(b)
(a)
(b)
(c)
Theorem 3. Let be the formal derivative associated with the Fibonacci grammar. For ,and
Proof. Equation (19) can be deduced from (14) by setting .
As for (20), we haveApplying the relation times repeatedly, we obtainwhich implies (20).
By setting in (19), we get the following corollary as a grammatical expression about the Fibonacci numbers.
Corollary 4. For , it holds that
As an application of Fibonacci grammar, we give a grammatical proof of Binet’s formula.
Theorem 5 (Binet’s formula). Let ,. For, it holds that
Equivalently,
Proof. Let , then . Since and , one can verify thatThus,which implies thatSimilarly, we haveandCombining (30) and (32), we obtainBy setting in (33), we obtainNow, we complete the proof.
By using the same grammar, we can also generate Lucas numbers in a grammatical manner, whose proof is omitted.
Theorem 6. For ,
Following properties of Fibonacci and Lucas numbers are classic and useful in this paper. For the sake of completeness, we provide a grammatical proof.
Lemma 7. For and ,
Proof. Notice thatNow, setting implies (36) by (23)–(25).
We have the equationwhich implies (37) by setting .
Finally, (38) can be deduced fromby setting .
Lemma 8. For ,
Proof. From (20), it holds thatActing the operator on the two hand sides of the above equation, we obtainwhich is changed toby setting . This is the special case of (42) in .
Notice thatThe last equation holds from (36). Setting in the above relation, we obtainwhich equals the right hand side of (42) by (45).
Lemma 9. For ,
Proof. We can verify thatSinceit holds thatwhich is reduced to (48) by setting .
Similarly, equations (49) and (50) can be obtained by simplifying the expressionand the expressionin the same manner as above and then setting .
4. -th Order Fibonacci Grammar, Binomial Fibonacci, and Lucas Identities
In this section, we provide a framework to prove the identities involving Fibonacci and Lucas numbers associated with binomial coefficients.
Let be a context-free grammar with an alphabet , and let be a constant for . We define the product of the grammar and to be the context-free grammar in which each letter corresponds the substitution rule , denoted as . A grammar on an alphabet is defined to be linear if for each letter , is a linear function on .
Lemma 10. Let be a linear context-free grammar with an alphabet , and let be a constant for . For and each linear function , it holds that
Proof. Because of the linearity of the operator , it is enough to show (56) holds for every letter . We prove the assertion by induction on . The case for is evident. Assume that the assertion holds for . Now, consider the case for , sincewhich equals . This completes the proof.
Consider the following context-free grammar :We call the -th order Fibonacci grammar. Let be the formal derivative associated with the grammar . According to (56), is equivalent to when acting on a linear functions of and . When , 1-th order Fibonacci grammar is just the Fibonacci grammar.
From (19), it can be verified thatThus,It should be noticed that is not equivalent to . For example,yetLetOne can easy to check thatFollowing assertion is critical for the proof.
Lemma 11. For , it holds that
Proof. We can verify thatSimilarly, it holds that . According to Lemma 10,where is the formal derivative associated with the grammarNotice that is as same as the -th order Fibonacci grammar by setting and . Thus,Thus,As for , one can verify thatThe last equation holds from (38). This completes the proof.
Now, we begin to proof binomial convolutions about Fibonacci and Lucas numbers.
Theorem 12. For , we have
Proof. According to (56),This deduces that the left hand side of (73) equals the summation:which equals by (65). This completes the proof of (73).
The left hand side of (74) can be calculated by a similar manner. We haveBy using Gaussian elimination, can be represented as the linear combination of and , namely,Thus,So,This completes the proof of (74).
As for (75), we need consider the following grammatical convolution It follows by Leibnitz formula thatwhich reduces by setting to beNow, let us calculate By using Gaussian elimination, can be represented as the linear combination of and , namely,Thus,The last equation holds from (38). This completes the proof.
Equations (74) and (75) are given by Church and Bicknell [3]. As far as we know, (73) is new.
Theorem 13. For and , we have
Proof. According to (19),Thus,Let denote the context-free grammarand let be the formal derivative associated with . Then,Next, let us turn to calculate and . Let , It can be verified thatAccording to (48)–(50), we haveAccording to (56),where is the formal derivative associated with the following grammarNotice that is as same as -th Fibonacci grammar by substituting into , respectively. Thus,This deduces thatThe last equation holds from (36). This completes the proof.
Theorem 14. For and , we haveand
Proof. Consider the grammar :Then, it can be easily see thatAccording to Leibnitz formula, the left hand side of (97) can be obtained from by setting . Now, the proof of (97) can be reduced to the calculation of .
Let . One can check thatIn (42), by setting and , we obtainAnd by setting and , we obtainThus,Then, can be viewed as the formal derivative associated with with the alphabet . According to (56),Thus,which equals by (36). This complete the proof of (97).
As for (98), we consider the following Leibnitz relation:Now, (98) can be obtained from (107) by setting sincewhich equals from (37). This complete the proof of (98).
Remark 15. It is easily to see that the technique of proving identities by using the simple Fibonacci grammar can be extended to study corresponding binomial convolutions involving Fibonacci polynomials , just considering the Fibonacci grammar. Meanwhile, one can extend the Fibonacci grammar to beto get generalized Fibonacci and generalized Lucas numbers, who are defined as the linear recurrenceand the initial conditions and .
Besides, the identities proved here are all from Leibnitz formula, hence in formTherefore, there are many identities involving Fibonacci and Lucas numbers who are not in the standard binomial form. For example, Kilic and Tasdemir [9] provided several binomial double summations in the formas well as the alternating binomial double summations in the formfor . It’s an interesting question to find a universal grammatical proof of these relations.
Data Availability
No underlying data were collected or produced in this study.
Conflicts of Interest
The author declares that there are no conflicts of interest.
Acknowledgments
This work was supported by the National Science Foundation of China (grant 12001404).