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Vieta-Pell and Vieta-Pell-Lucas polynomials
Advances in Difference Equations volume 2013, Article number: 224 (2013)
In the present paper, we introduce the recurrence relation of Vieta-Pell and Vieta-Pell-Lucas polynomials. We obtain the Binet form and generating functions of Vieta-Pell and Vieta-Pell-Lucas polynomials and define their associated sequences. Moreover, we present some differentiation rules and finite summation formulas.
Andre-Jeannin  introduced a class of polynomials defined by
with the initial values and .
Vieta-Lucas polynomials were studied as Vieta polynomials by Robbins . Vieta-Fibonacci and Vieta-Lucas polynomials are defined by
Chebyshev polynomials are a sequence of orthogonal polynomials which can be defined recursively. Recall that the n th Chebyshev polynomials of the first kind and second kind are denoted by and , respectively.
It is well known that the Chebyshev polynomials of the first kind and second kind are closely related to Vieta-Fibonacci and Vieta-Lucas polynomials. So, in  Vitula and Slota redefined Vieta polynomials as modified Chebyshev polynomials. The related features of Vieta and Chebyshev polynomials are given as
For , we consider and polynomials by the following recurrence relations:
where , and , . We call the n th Vieta-Pell polynomial and the n th Vieta-Pell-Lucas polynomial.
The relations below are obvious
The first few terms of and are as follows:
The aim of this paper is to determine the recursive key features of Vieta-Pell and Vieta-Pell-Lucas polynomials. In conjunction with these properties, we examine their interrelations and define their associated sequences. Furthermore, we present some differentiation rules and summation formulas.
2 Main results
Some fundamental recursive properties of Vieta-Pell and Vieta Pell-Lucas polynomials are given in this section.
Vieta-Pell and Vieta-Pell-Lucas polynomials have the following characteristic equation:
with the roots α and β
Also, α and β satisfy the following equations:
By appropriate procedure, we can easily find the Binet forms as
Vieta-Pell and Vieta-Pell-Lucas polynomials can be defined by the following generating functions:
We can also extend the definition of and to the negative index
can be written like the coefficients of Pell polynomials in . Moreover,
For example, for we can find
Let denote the n th Pell number, so we have
2.1 Interrelations of and
Most of the equations below can be obtained by using the Binet form and convenient routine operations
Proposition 1 .
Proof Consider the expression . Then α, β, Δ are replaced by , , , respectively. So, , , and by using the Binet form, the proof is completed. □
2.2 Associated sequences
Definition 1 The k th associated sequences and of and are defined by, respectively ()
where and .
are the members of the first associated sequences and . If (6) and (7) are applied repeatedly, the results emerge
Some special values of and
2.3 Differentiation formulas
Since the derivation function of is a polynomial, all of the derivatives must exist for all real numbers. Thus, we can give the following formulas.
Proof If we take the limit on , we have the numerical value of at and .
Since , , apply L’Hôpital’s rule:
So, the proof for is similar. □
2.4 Some summation formulas
In this section we deal with the matrix
By induction, we have
So, the matrix V generates Vieta-Pell and Vieta-Pell-Lucas polynomials. Hence,
and from (1.10) in , we get
It is known that
From (8) and (11), the elementary formulas for are obvious
If we use the matrix technique for summation in , we get the first finite summation as follows.
Proof (i) Let the matrix A,
be the series of matrices. Then we have
Thus, the proof is completed.
This completes the proof. □
Theorem 1 Let V be a square matrix such that . Then, for all ,
where is the nth Vieta-Pell polynomial and I is a unit matrix.
Proof The proof is obvious from induction. □
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The authors thank the referees for their valuable suggestions, which improved the standard of the paper.
The authors declare that they have no competing interests.
The authors declare that the research was realized in collaboration with the same responsibility and contributions. Both authors read and approved the final manuscript.