Abstract
In this work, we bring to light the properties of newly formed polynomial sequences at each vertex of Pell polynomial sequences placed clockwise at each vertex in the n-gon. We compute the relation among the polynomials with such vertices. Moreover, in an n-gon, we generate a recurrence relation for a sequence giving the mth term formed at the kth. Similar to the situations we are talking about, we applied to Pell-Lucas polynomials, Jacobsthal polynomials, and Jacobsthal-Lucas polynomials and have obtained new relations recurrence.
1. Introduction
It is an interesting tool for many researchers to know about number sequences, especially thrust is given to Fibonacci numbers. In this direction, many researchers work on it and study its wings, and the Lucas numbers became a vital role for them to study as can be seen in [1–8]. While framing these sequences, other sequences came into existence such as Pell, Pell-Lucas, Jacobsthal and Jacobsthal-Lucas by replacing the initial conditions [9–15].
As in [16–23], in engineering, science, and technology, the structure of such sequences with generalizations plays a vital significance art and architecture. In studying such discipline, Pell, Pell-Lucas, Jacobsthal, and Jacobsthal-Lucas polynomials have shown their significance as in [24–27].
Recently, the study of triangles with coordinates of vertices have been studied in [14, 19].
We have made a new study for polynomials of Pell, Pell-Lucas, Jacobsthal, and Jacobsthal by reference to [19] articles in this study.
The motivation of this work is to study the clockwise nature of the Pell, Pell-Lucas, Jacobsthal, and Jacobsthal-Lucas polynomials to be located on the vertices of the polygons.
We have computed some interesting relations in n-gon concerning the vertex Bk corresponding to th term for . Also, a relation corresponding to a vertex among the numbers will be determined.
2. Materials and Methods
Definition 1. The polynomials of the given by linear recurrence relationwith and called as Pell polynomials.
The negative Pell polynomials are defined by the recurrence relation,The first few terms of this sequence are .
For Fibonacci sequence, the characteristic equation isThe roots of this equation are and .The Binet formula of this sequence is
Definition 2. The linear reoccurrence relation for Pell-Lucas polynomials iswith and .
The first few terms of this sequence are .
The negative Pell-Lucas polynomials are defined by the recurrence relation,The Binet formula of this sequence is
Definition 3. The recurrence relation for Jacobsthal polynomials iswith and .
Note thatThe first few terms of this sequence are .
The characteristic equation for the Jacobsthal polynomials sequence isThe roots of this equation are and ,The Binet formula of this sequence is
Definition 4. The recurrence relation for Jacobsthal-Lucas polynomials iswith and .
Note thatThe first few terms of this sequence are .
The Binet formula of this sequence is
3. Results and Discussion
3.1. New Recurrence Relations for the Pell and Pell-Lucas Numbers
From here onwards, we shall consider a continuously clockwise sense approach to the polygon of the Pell polynomials and Pell-Lucas polynomials. We will continue to n-gon starting this placement from a point. For such a polygon, let us call each vertex point as for . Here, we aim at finding a relationship that locates any term of new Bk sequences to every vertex of the polygon. We establish such a relationship in Theorem 3.1.1 to Pell polynomials and in Theorem 3.1.3 to Pell-Lucas polynomials.
Let the vertex number be represented by and represents the order of any term of the new sequence occurring at any vertex in any n-gon, with and .
Taking , the Pell polynomials in succession over a dot are given, as depicted in Figure 1.
For yields and giving a recurrence relation,with and of the form of Pell polynomials.
For , choose the Pell polynomials in succession at the endpoints of a line, as depicted in Figure 2.
So, it determines a recurrence relation of the form
For , the vertices of the triangle will be labeled by Pell polynomials clockwise, as given in Figure 3.
So, it determines a recurrence relation of the form
Now, for , the Pell numbers are taken clockwise on the vertices of a tetragon, as given in Figure 4.
So, it determines a recurrence relation of the form
Similarly, for an n-gon, we take clockwise the Pell numbers on its vertices, as given in Figure 5.
For example, in a new sequence to determine the second term of it occurring at the vertex in a 3-gon.
Thus, the value for , , and corresponds to the 3rd term of sequence in Figure 3.
This term must be .
Now, to determine this value from the recurrence relation with fromso that
In Figure 5, we define a new sequence created at the vertex .
. As is the first term to this sequence, the second term is and the third term is , and if it continues like this, it is observed that the mth term will be of the new sequence formed in an n-gon.
In addition, the coefficient of is Pell-Lucas polynomials and is the coefficient of .
So, we have the following result.
Theorem 1. For , the relationis satisfied, giving mth term of the sequence with respect to vertex .
Proof. Using Binet formulas of Pell and Pell-Lucas polynomials, we seeBecause of , we haveLet us indicate the coefficients corresponding to this sequence in Table 1.
Table 1 clearly indicates that the first components are Pell-Lucas polynomials, .
Table 1 clearly indicates that the second are {, , , , , , , , }. {, , , , , , , , } is numbered as sequences A033999 in the On-Line Encyclopedia of Integer Sequences [25].
Theorem 2. The relation for between the numbers with respect to the vertex is
Proof. To establish this result, we induct on .
Thus, when , we seeFor , we assume the claim is true, that is,For , let us show it is also true.
For this, if we use (23), then we seeBy using (28), we haveWhen , we haveThe equality has been shown for .
Now, when , we getThus, the proof is obtained.
Let us give example as an application. For , and , we haveNow, let us place the Pell-Lucas polynomials like the Pell polynomials continuously clockwise at the vertices of a regular polygon.
For this new sequence, the th term with respect to the vertex Bk of an n-gon is .
So, we have the following theorem about this situation.
Theorem 3. If on the vertices of an n-gon, the Pell-Lucas polynomials are placed clockwise, then for , the mth term of sequence with respect to the vertex is
Proof. To prove the result, by employing Binet formula for Pell-Lucas polynomials, we seeBecause of , we have
Theorem 4. For an n-gon, if Pell-Lucas polynomials are kept on the vertices clockwise sense. Then, for , there exists a relation of the formbetween the numbers with respect to the vertex
Proof. The result can be proved in a similar fashion as Theorem 3.1.2.
For example, when and , we have
3.2. Recurrences Notion Corresponding to Jacobsthal and Jacobsthal-Lucas Polynomials
In this direction, we involve the clockwise sense of the Jacobsthal polynomials on the point, at extreme points, at vertices of the triangle, and lastly at the vertices of an n-gon.
Since for this new sequence, the mth term computed at vertex is for an -gon. Also, the coefficient of is Jacobsthal-Lucas polynomials and the coefficient of is .
In this case, we have the following result.
Theorem 5. On an n-gon, suppose Jacobsthal numbers are kept on its vertices clockwise. Then, for , we haveand it determines the mth term of the sequence with respect to vertex .
Proof. By using the Binet formulas of Jacobsthal and Jacobsthal-Lucas polynomials, we have the following:Since , , thusIn Table 2, we write the coefficients concerning the above sequences.
So, clearly in Table 2, the Jacobsthal-Lucas polynomials are first components.
The second components in Table 2 are {, , , }. {, , , , , , , , } is enumerate as sequences A084633 in the On-Line Encyclopedia of Integer Sequences [25].
Theorem 6. On an n-gon, let the Pell-Lucas polynomials be kept on its vertices clockwise. Then, for , there exists a relation corresponding to numbers with respect to the vertex as
Proof. As in Theorem 2, the proof can be performed.
Now, for an -gon, taking the Jacobsthal-Lucas polynomials on the vertices clockwise and in this n-gon, the mth term formed to this new sequence at the vertex is .
We have the following result in this direction.
Theorem 7. On an n-gon, if the Jacobsthal-Lucas polynomials are kept on their vertices clockwise. Then, for , the mth term with respect to the vertex is
Proof. To establish this result, Binet formula for Jacobsthal-Lucas numbers is employed and givesSince , , thus
Theorem 8. On an n-gon, let the Jacobsthal-Lucas numbers be kept on its vertices clockwise. Then, for , the numbers with respect to the vertex areProof. The result can be proved in a similar fashion as Theorem 2.
For example, when and , we get
4. Conclusions
In this paper, we have described the clockwise sense of Pell, Pell-Lucas, Jacobsthal, and Jacobsthal-Lucas polynomials on the vertices of polygons. We have computed the recurrence relations for numbers by sorting the Pell polynomials around the polygons. Also, we have determined new correlations between the terms with respect to each vertex and generated new findings by employing these results. The work can be further extended to Jacobsthal and Jacobsthal-Lucas numbers and Fibonacci and Lucas numbers. [20-27]
Data Availability
Data sharing is not applicable to this article as no data sets were generated or analyzed during the current study
Conflicts of Interest
The authors declare that there are no conflicts of interest.