Vieta-Pell and Vieta-Pell-Lucas polynomials
© Tasci and Yalcin; licensee Springer 2013
Received: 4 April 2013
Accepted: 11 July 2013
Published: 24 July 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.
with the initial values and .
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.
where , and , . We call the n th Vieta-Pell polynomial and the n th Vieta-Pell-Lucas polynomial.
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.
The first ten coefficients of
2.1 Interrelations of and
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
where and .
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.
So, the proof for is similar. □
2.4 Some summation formulas
If we use the matrix technique for summation in , we get the first finite summation as follows.
This completes the proof. □
where is the nth Vieta-Pell polynomial and I is a unit matrix.
Proof The proof is obvious from induction. □
The authors thank the referees for their valuable suggestions, which improved the standard of the paper.
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