International Scholarly Research Notices

International Scholarly Research Notices / 2011 / Article

Research Article | Open Access

Volume 2011 |Article ID 674167 | https://doi.org/10.5402/2011/674167

Tian-Xiao He, Peter J.-S. Shiue, Tsui-Wei Weng, "Sequences of Numbers Meet the Generalized Gegenbauer-Humbert Polynomials", International Scholarly Research Notices, vol. 2011, Article ID 674167, 16 pages, 2011. https://doi.org/10.5402/2011/674167

Sequences of Numbers Meet the Generalized Gegenbauer-Humbert Polynomials

Academic Editor: W. Liu
Received06 Jul 2011
Accepted25 Aug 2011
Published26 Oct 2011

Abstract

Here we present a connection between a sequence of numbers generated by a linear recurrence relation of order 2 and sequences of the generalized Gegenbauer-Humbert polynomials. Many new and known formulas of the Fibonacci, the Lucas, the Pell, and the Jacobsthal numbers in terms of the generalized Gegenbauer-Humbert polynomial values are given. The applications of the relationship to the construction of identities of number and polynomial value sequences defined by linear recurrence relations are also discussed.

1. Introduction

Many number and polynomial sequences can be defined, characterized, evaluated, and/or classified by linear recurrence relations with certain orders. A number sequence {π‘Žπ‘›} is called sequence of order 2 if it satisfies the linear recurrence relation of order 2: π‘Žπ‘›=π‘π‘Žπ‘›βˆ’1+π‘žπ‘Žπ‘›βˆ’2,𝑛β‰₯2,(1.1) for some nonzero constants 𝑝 and π‘ž and initial conditions π‘Ž0 and π‘Ž1. In Mansour [1], the sequence {π‘Žπ‘›}𝑛β‰₯0 defined by (1.1) is called Horadam’s sequence, which was introduced in 1965 by Horadam [2]. In [1] also the generating functions for powers of Horadam’s sequence are obtained. To construct an explicit formula of its general term, one may use a generating function, characteristic equation, or a matrix method (see Comtet [3], Hsu [4], Strang [5], Wilf [6], etc.) In [7], Benjamin and Quinn presented many elegant combinatorial meanings of the sequence defined by recurrence relation (1.1). For instance, π‘Žπ‘› counts the number of ways to tile an 𝑛-board (i.e., board of length 𝑛) with squares (representing 1s) and dominoes (representing 2s) where each tile, except the initial one, has a color. In addition, there are 𝑝 colors for squares and π‘ž colors for dominoes. In particular, Aharonov et al. (see [8]) have proved that the solution of any sequence of numbers that satisfies a recurrence relation of order 2 with constant coefficients and initial conditions π‘Ž0=0 and π‘Ž1=1, called the primary solution, can be expressed in terms of the Chebyshev polynomial values. For instance, the authors show 𝐹𝑛=π‘–βˆ’π‘›π‘ˆπ‘›(𝑖/2) and 𝐿𝑛=2π‘–βˆ’π‘›π‘‡π‘›(𝑖/2), where 𝐹𝑛 and 𝐿𝑛, respectively, are the Fibonacci numbers and Lucas numbers, and 𝑇𝑛 and π‘ˆπ‘› are the Chebyshev polynomials of the first kind and the second kind, respectively. Some identities drawn from those relations were given by Beardon in [9]. Marr and Vineyard in [10] use the relationship to establish explicit expression of five-diagonal Toeplitz determinants. In [11], the first two authors presented a new method to construct an explicit formula of {π‘Žπ‘›} generated by (1.1). For the sake of the reader's convenience, we cite this result as follows.

Proposition 1.1 (see [11]). Let {π‘Žπ‘›} be a sequence of order 2 satisfying linear recurrence relation (1.1), and let 𝛼 and 𝛽 be two roots of of quadratic equation π‘₯2βˆ’π‘π‘₯βˆ’π‘ž=0. Then π‘Žπ‘›=⎧βŽͺ⎨βŽͺβŽ©ξ‚΅π‘Ž1βˆ’π›½π‘Ž0ξ‚Άπ›Όπ›Όβˆ’π›½π‘›βˆ’ξ‚΅π‘Ž1βˆ’π›Όπ‘Ž0ξ‚Άπ›½π›Όβˆ’π›½π‘›,if𝛼≠𝛽,π‘›π‘Ž1π›Όπ‘›βˆ’1βˆ’(π‘›βˆ’1)π‘Ž0𝛼𝑛,if𝛼=𝛽.(1.2)

A sequence of the generalized Gegenbauer-Humbert polynomials {π‘ƒπ‘›πœ†,𝑦,𝐢(π‘₯)}𝑛β‰₯0 is defined by the expansion (see, e.g., Comtet [3], Gould [12], Lidl et al. [13], the two authors with He et al. [14]) ξ€·Ξ¦(𝑑)β‰‘πΆβˆ’2π‘₯𝑑+𝑦𝑑2ξ€Έβˆ’πœ†=𝑛β‰₯0π‘ƒπ‘›πœ†,𝑦,𝐢(π‘₯)𝑑𝑛,(1.3) where πœ†>0, 𝑦 and 𝐢≠0 are real numbers. As special cases of (1.3), we consider π‘ƒπ‘›πœ†,𝑦,𝐢(π‘₯) as follows (see [14]): 𝑃𝑛1,1,1(π‘₯)=π‘ˆπ‘›(π‘₯), the Chebyshev polynomial of the second kind,𝑃𝑛1/2,1,1(π‘₯)=πœ“π‘›(π‘₯), the Legendre polynomial,𝑃𝑛1,βˆ’1,1(π‘₯)=𝑃𝑛+1(π‘₯), the Pell polynomial,𝑃𝑛1,βˆ’1,1(π‘₯/2)=𝐹𝑛+1(π‘₯), the Fibonacci polynomial,𝑃𝑛1,2,1(π‘₯/2)=Φ𝑛+1(π‘₯), the Fermat polynomial of the first kind,𝑃𝑛1,2π‘Ž,2(π‘₯)=𝐷𝑛(π‘₯,π‘Ž), the Dickson polynomial of the second kind, π‘Žβ‰ 0 (see, e.g., [13]),

where π‘Ž is a real parameter, and 𝐹𝑛=𝐹𝑛(1) is the Fibonacci number. In particular, if 𝑦=𝐢=1, the corresponding polynomials are called the Gegenbauer polynomials (see [3]). More results on the Gegenbauer-Humbert-type polynomials can be found in [15] by Hsu and in [16] by the second author and Hsu, and so forth.

Similarly, for a class of the generalized Gegenbauer-Humbert polynomial sequences defined by π‘ƒπ‘›πœ†,𝑦,𝐢(π‘₯)=2π‘₯πœ†+π‘›βˆ’1π‘ƒπΆπ‘›πœ†,𝑦,πΆπ‘›βˆ’1(π‘₯)βˆ’π‘¦2πœ†+π‘›βˆ’2π‘ƒπΆπ‘›πœ†,𝑦,πΆπ‘›βˆ’2(π‘₯),(1.4) for all 𝑛β‰₯2 with initial conditions𝑃0πœ†,𝑦,𝐢(π‘₯)=Ξ¦(0)=πΆβˆ’πœ†,𝑃1πœ†,𝑦,𝐢(π‘₯)=Ξ¦ξ…ž(0)=2πœ†π‘₯πΆβˆ’πœ†βˆ’1,(1.5) the following theorem has been obtained in [11].

Theorem 1.2 (see [11]). Let √π‘₯≠±𝐢𝑦. The generalized Gegenbauer-Humbert polynomials {𝑃𝑛1,𝑦,𝐢(π‘₯)}𝑛β‰₯0 defined by expansion (1.3) can be expressed as 𝑃𝑛1,𝑦,𝐢(π‘₯)=πΆβˆ’π‘›βˆ’2ξ‚€π‘₯+√π‘₯2ξ‚βˆ’πΆπ‘¦π‘›+1βˆ’ξ‚€π‘₯βˆ’βˆšπ‘₯2ξ‚βˆ’πΆπ‘¦π‘›+12√π‘₯2βˆ’πΆπ‘¦.(1.6)

In this paper, we will use an alternative form of (1.2) to establish a relationship between the number sequences defined by recurrence relation (1.1) and the generalized Gegenbauer-Humbert polynomial sequences defined by (1.4). Our results are suitable for all such number sequences defined by (1.1) with arbitrary initial conditions π‘Ž0 and π‘Ž1, which includes the results in [8, 9] as our special cases. Many new and known formulas of the Fibonacci, the Lucas, the Pell, and the Jacobsthal numbers in terms of the generalized Gegenbauer-Humbert polynomial values and applications of the established relationship to the construction of identities of number and polynomial value sequences will be presented in Section 3.

2. Main Results

We now modify the explicit formula of the number sequences defined by linear recurrence relations of order 2. If 𝛼≠𝛽, the first formula in (1.2) can be written as π‘Žπ‘›=π‘Ž1(π›Όπ‘›βˆ’π›½π‘›)βˆ’π‘Ž0ξ€·π›Όπ›Όπ›½π‘›βˆ’1βˆ’π›½π‘›βˆ’1ξ€Έ=π‘Žπ›Όβˆ’π›½1(π›Όπ‘›βˆ’π›½π‘›)+π‘Ž0π‘žξ€·π›Όπ‘›βˆ’1βˆ’π›½π‘›βˆ’1ξ€Έ,π›Όβˆ’π›½(2.1) where the last step is due to 𝛼 and 𝛽 being solutions of 𝑑2βˆ’π‘π‘‘βˆ’π‘ž=0. Noting that 𝛼2βˆ’π‘π›Ό=𝛼2βˆ’(𝛼+𝛽)𝛼=βˆ’π›Όπ›½=π‘ž and 𝛼(π›Όβˆ’π‘)=βˆ’π›Όπ›½=𝛽(π›½βˆ’π‘), we may further write the above last expression of π‘Žπ‘› as π‘Žπ‘›=π‘Ž1(π›Όπ‘›βˆ’π›½π‘›)+π‘Ž0𝛼2π›Όβˆ’π‘π›Όξ€Έξ€·π‘›βˆ’1βˆ’π›½π‘›βˆ’1ξ€Έ=π‘Žπ›Όβˆ’π›½1(π›Όπ‘›βˆ’π›½π‘›)+π‘Ž0𝛼2ξ€Έπ›Όβˆ’π‘π›Όπ‘›βˆ’1βˆ’π‘Ž0𝛽2ξ€Έπ›½βˆ’π‘π›½π‘›βˆ’1=π‘Žπ›Όβˆ’π›½0𝛼𝑛+1βˆ’π›½π‘›+1ξ€Έ+ξ€·π‘Ž1βˆ’π‘Ž0𝑝(π›Όπ‘›βˆ’π›½π‘›).π›Όβˆ’π›½(2.2)

Denote π‘Ÿ(π‘₯)=π‘₯+√π‘₯2βˆ’πΆπ‘¦ and 𝑠(π‘₯)=π‘₯βˆ’βˆšπ‘₯2βˆ’πΆπ‘¦. Comparing expressions (2.2) and (1.6), we have reason to consider the following transform: for a nonzero real or complex number π‘˜, we set π›ΌβˆΆ=π‘Ÿ(π‘₯)π‘˜,π›½βˆΆ=𝑠(π‘₯)π‘˜(2.3) for a certain π‘₯ depending on 𝛼, 𝛽, and π‘˜, which we will find out later. Denote 𝛼+𝛽=𝑝 and 𝛼𝛽=βˆ’π‘ž; that is, 𝛼 and 𝛽 are roots of 𝑑2βˆ’π‘π‘‘βˆ’π‘ž. By adding the two equations in (2.3) side by side, we obtain 2π‘₯=π‘˜π‘. Thus, when π‘₯=π‘˜π‘/2, the equations in (2.2) hold. Meanwhile, by using (π›Όβˆ’π›½)2=(𝛼+𝛽)2βˆ’4𝛼𝛽=𝑝2+4π‘ž, we have π‘Ÿβˆš(π‘₯)βˆ’π‘ (π‘₯)=2π‘₯2βˆšβˆ’πΆπ‘¦=π‘˜(π›Όβˆ’π›½)=π‘˜π‘2+4π‘ž,(2.4) where π‘₯=π‘˜π‘/2. Therefore, we obtain 2ξƒŽξ‚΅π‘˜π‘2ξ‚Ά2βˆšβˆ’πΆπ‘¦=π‘˜π‘2+4π‘ž,(2.5) which implies ξ‚™π‘˜=Β±πΆπ‘¦βˆ’π‘ž.(2.6) We first consider the case of βˆšπ‘˜=βˆ’πΆπ‘¦/π‘ž.

We now substitute π‘Ÿ(π‘₯)=π‘˜π›Ό, 𝑠(π‘₯)=π‘˜π›½, π‘₯=π‘˜π‘/2, and βˆšπ‘˜=βˆ’πΆπ‘¦/π‘ž into (2.2) and simplify as follows:π‘Žπ‘›=π‘Ž0ξ€·(π‘Ÿ(π‘₯)/π‘˜)𝑛+1βˆ’(𝑠(π‘₯)/π‘˜)𝑛+1ξ€Έ+ξ€·π‘Ž1βˆ’π‘Ž0𝑝((π‘Ÿ(π‘₯)/π‘˜)π‘›βˆ’(𝑠(π‘₯)/π‘˜)𝑛)=π‘Ž(1/π‘˜)(π‘Ÿ(π‘₯)βˆ’π‘ (π‘₯))0ξ€·π‘Ÿπ‘›+1(π‘₯)βˆ’π‘ π‘›+1ξ€Έξ€·π‘Ž(π‘₯)+π‘˜1βˆ’π‘Ž0𝑝(π‘Ÿπ‘›(π‘₯)βˆ’π‘ π‘›(π‘₯))π‘˜π‘›(π‘Ÿ(π‘₯)βˆ’π‘ (π‘₯))=π‘Ž0𝐢𝑛+2ξ‚΅ξ‚™βˆ’π‘žξ‚ΆπΆπ‘¦π‘›π‘ƒπ‘›1,𝑦,πΆξ‚΅π‘˜π‘2ξ‚Ά+ξ€·π‘Ž1βˆ’π‘Ž0𝑝𝐢𝑛+1ξ‚΅ξ‚™βˆ’π‘žξ‚ΆπΆπ‘¦π‘›βˆ’1𝑃1,𝑦,πΆπ‘›βˆ’1ξ‚΅π‘˜π‘2ξ‚Ά=π‘Ž0𝐢𝑛+2ξ‚΅ξ‚™βˆ’π‘žξ‚ΆπΆπ‘¦π‘›π‘ƒπ‘›1,𝑦,𝐢𝑝2𝐢𝑦ξƒͺ+ξ€·π‘Žβˆ’π‘ž1βˆ’π‘Ž0𝑝𝐢𝑛+1ξ‚΅ξ‚™βˆ’π‘žξ‚ΆπΆπ‘¦π‘›βˆ’1𝑃1,𝑦,πΆπ‘›βˆ’1𝑝2𝐢𝑦ξƒͺ.βˆ’π‘ž(2.7) Similarly, for βˆšπ‘˜=βˆ’βˆ’πΆπ‘¦/π‘ž, we have π‘Žπ‘›=π‘Ž0𝐢𝑛+2ξ‚΅βˆ’ξ‚™βˆ’π‘žξ‚ΆπΆπ‘¦π‘›π‘ƒπ‘›1,𝑦,πΆξƒ©βˆ’π‘2𝐢𝑦ξƒͺ+ξ€·π‘Žβˆ’π‘ž1βˆ’π‘Ž0𝑝𝐢𝑛+1ξ‚΅βˆ’ξ‚™βˆ’π‘žξ‚ΆπΆπ‘¦π‘›βˆ’1𝑃1,𝑦,πΆπ‘›βˆ’1ξƒ©βˆ’π‘2𝐢𝑦ξƒͺ.βˆ’π‘ž(2.8)

Therefore, we obtain our main result.

Theorem 2.1. Let sequence {π‘Žπ‘›} be defined by π‘Žπ‘›=π‘π‘Žπ‘›βˆ’1+π‘žπ‘Žπ‘›βˆ’2 (𝑛β‰₯2) with initial conditions π‘Ž0 and π‘Ž1. Then, π‘Žπ‘› can be presented as (2.7) and (2.8). In particular, for (𝑦,𝐢)=(1,1),(βˆ’1,1),(2,1), and (2π‘Ž,2)(π‘Žβ‰ 0), respectively, one has π‘Žπ‘›=π‘Ž0ξ‚€βˆšξ‚βˆ’π‘žπ‘›π‘ˆπ‘›ξƒ©π‘2√ξƒͺ+ξ€·π‘Žβˆ’π‘ž1βˆ’π‘Ž0π‘ξ€Έξ‚€βˆšξ‚βˆ’π‘žπ‘›βˆ’1π‘ˆπ‘›βˆ’1𝑝2√ξƒͺ,π‘Žβˆ’π‘žπ‘›=π‘Ž0ξ‚€βˆšπ‘žξ‚π‘›π‘ƒπ‘›+1𝑝2βˆšπ‘žξƒͺ+ξ€·π‘Ž1βˆ’π‘Ž0π‘ξ€Έξ‚€βˆšπ‘žξ‚π‘›βˆ’1𝑃𝑛𝑝2βˆšπ‘žξƒͺ,π‘Žπ‘›=π‘Ž0ξ‚€βˆšπ‘žξ‚π‘›πΉπ‘›+1ξƒ©π‘βˆšπ‘žξƒͺ+ξ€·π‘Ž1βˆ’π‘Ž0π‘ξ€Έξ‚€βˆšπ‘žξ‚π‘›βˆ’1πΉπ‘›ξƒ©π‘βˆšπ‘žξƒͺ,π‘Žπ‘›=π‘Ž0ξƒ©ξ‚™βˆ’π‘ž2ξƒͺ𝑛Φ𝑛+1𝑝2ξ‚Ά+ξ€·π‘Žβˆ’π‘ž1βˆ’π‘Ž0π‘ξ€Έξƒ©ξ‚™βˆ’π‘ž2ξƒͺπ‘›βˆ’1Φ𝑛𝑝2ξ‚Ά,π‘Žβˆ’π‘žπ‘›=π‘Ž02𝑛+2ξƒ©ξ‚™βˆ’π‘žξƒͺ4π‘Žπ‘›π·π‘›ξ‚΅π‘ξ‚™π‘Žξ‚Ά+ξ€·π‘Žβˆ’π‘ž,π‘Ž1βˆ’π‘Ž0𝑝2𝑛+1ξƒ©ξ‚™βˆ’π‘žξƒͺ4π‘Žπ‘›βˆ’1π·π‘›βˆ’1ξ‚΅π‘ξ‚™π‘Žξ‚Ά,π‘Žβˆ’π‘ž,π‘Žπ‘›=π‘Ž0ξ‚€βˆ’βˆšξ‚βˆ’π‘žπ‘›π‘ˆπ‘›ξƒ©βˆ’π‘2√ξƒͺ+ξ€·π‘Žβˆ’π‘ž1βˆ’a0π‘ξ€Έξ‚€βˆ’βˆšξ‚βˆ’π‘žπ‘›βˆ’1π‘ˆπ‘›βˆ’1ξƒ©βˆ’π‘2√ξƒͺ,π‘Žβˆ’π‘žπ‘›=π‘Ž0ξ‚€βˆ’βˆšπ‘žξ‚π‘›π‘ƒπ‘›+1ξƒ©βˆ’π‘2βˆšπ‘žξƒͺ+ξ€·π‘Ž1βˆ’π‘Ž0π‘ξ€Έξ‚€βˆ’βˆšπ‘žξ‚π‘›βˆ’1π‘ƒπ‘›ξƒ©βˆ’π‘2βˆšπ‘žξƒͺ,π‘Žπ‘›=π‘Ž0ξ‚€βˆ’βˆšπ‘žξ‚π‘›πΉπ‘›+1ξƒ©βˆ’π‘βˆšπ‘žξƒͺ+ξ€·π‘Ž1βˆ’π‘Ž0π‘ξ€Έξ‚€βˆ’βˆšπ‘žξ‚π‘›βˆ’1πΉπ‘›ξƒ©βˆ’π‘βˆšπ‘žξƒͺ,π‘Žπ‘›=π‘Ž0ξƒ©βˆ’ξ‚™βˆ’π‘ž2ξƒͺ𝑛Φ𝑛+1ξ‚΅ξ‚™βˆ’π‘2ξ‚Ά+ξ€·π‘Žβˆ’π‘ž1βˆ’π‘Ž0π‘ξ€Έξƒ©βˆ’ξ‚™βˆ’π‘ž2ξƒͺπ‘›βˆ’1Ξ¦π‘›ξ‚΅ξ‚™βˆ’π‘2ξ‚Ά,π‘Žβˆ’π‘žπ‘›=π‘Ž02𝑛+2ξƒ©βˆ’ξ‚™βˆ’π‘žξƒͺ4π‘Žπ‘›π·π‘›ξ‚΅ξ‚™βˆ’π‘π‘Žξ‚Ά+ξ€·π‘Žβˆ’π‘ž,π‘Ž1βˆ’π‘Ž0𝑝2𝑛+1ξƒ©βˆ’ξ‚™βˆ’π‘žξƒͺ4π‘Žπ‘›βˆ’1π·π‘›βˆ’1ξ‚΅ξ‚™βˆ’π‘π‘Žξ‚Ά,βˆ’π‘ž,π‘Ž(2.9) where π‘ˆπ‘›(π‘₯), 𝑃𝑛(π‘₯), 𝐹𝑛(π‘₯), Φ𝑛(π‘₯), and 𝐷𝑛(π‘₯,π‘Ž) are the 𝑛th degree Chebyshev polynomial of the second kind, the Pell polynomial, the Fibonacci polynomial, the Fermat polynomial, and the Dickson polynomial of the second kind, respectively.

For the special cases of π‘Ž0 and π‘Ž1, we have the following corollaries.

Corollary 2.2. Let sequence {π‘Žπ‘›} be defined by π‘Žπ‘›=π‘π‘Žπ‘›βˆ’1+π‘žπ‘Žπ‘›βˆ’2 (𝑛β‰₯2) with initial conditions π‘Ž0=0 and π‘Ž1=𝑑. Then π‘Žπ‘›ξ‚€βˆš=π‘‘ξ‚βˆ’π‘žπ‘›βˆ’1π‘ˆπ‘›βˆ’1𝑝2√ξƒͺ,π‘Žβˆ’π‘žπ‘›ξ‚€βˆš=π‘‘π‘žξ‚π‘›βˆ’1𝑃𝑛𝑝2βˆšπ‘žξƒͺ,π‘Žπ‘›ξ‚€βˆš=π‘‘π‘žξ‚π‘›βˆ’1πΉπ‘›ξƒ©π‘βˆšπ‘žξƒͺ,π‘Žπ‘›ξƒ©ξ‚™=π‘‘βˆ’π‘ž2ξƒͺπ‘›βˆ’1Φ𝑛𝑝2ξ‚Ά,π‘Žβˆ’π‘žπ‘›=𝑑2𝑛+1ξƒ©ξ‚™βˆ’π‘žξƒͺ4π‘Žπ‘›βˆ’1π·π‘›βˆ’1ξ‚΅π‘ξ‚™π‘Žξ‚Ά,π‘Žβˆ’π‘ž,π‘Žπ‘›ξ‚€βˆ’βˆš=π‘‘ξ‚βˆ’π‘žπ‘›βˆ’1π‘ˆπ‘›βˆ’1ξƒ©βˆ’π‘2√ξƒͺ,π‘Žβˆ’π‘žπ‘›ξ‚€βˆ’βˆš=π‘‘π‘žξ‚π‘›βˆ’1π‘ƒπ‘›ξƒ©βˆ’π‘2βˆšπ‘žξƒͺ,π‘Žπ‘›ξ‚€βˆ’βˆš=π‘‘π‘žξ‚π‘›βˆ’1πΉπ‘›ξƒ©βˆ’π‘βˆšπ‘žξƒͺ,π‘Žπ‘›ξƒ©βˆ’ξ‚™=π‘‘βˆ’π‘ž2ξƒͺπ‘›βˆ’1Ξ¦π‘›ξ‚΅ξ‚™βˆ’π‘2ξ‚Ά,π‘Žβˆ’π‘žπ‘›=𝑑2𝑛+1ξƒ©βˆ’ξ‚™βˆ’π‘žξƒͺ4π‘Žπ‘›βˆ’1π·π‘›βˆ’1ξ‚΅ξ‚™βˆ’π‘π‘Žξ‚Ά.βˆ’π‘ž,π‘Ž(2.10)

Corollary 2.3. Let sequence {π‘Žπ‘›} be defined by π‘Žπ‘›=π‘π‘Žπ‘›βˆ’1+π‘žπ‘Žπ‘›βˆ’2 (𝑛β‰₯2) with initial conditions π‘Ž0=𝑐 and π‘Ž1=𝑝𝑐. Then π‘Žπ‘›ξ‚€βˆš=π‘ξ‚βˆ’π‘žπ‘›π‘ˆπ‘›ξƒ©π‘2√ξƒͺ,π‘Žβˆ’π‘žπ‘›ξ‚€βˆš=π‘π‘žξ‚π‘›π‘ƒπ‘›+1𝑝2βˆšπ‘žξƒͺ,π‘Žπ‘›ξ‚€βˆš=π‘π‘žξ‚π‘›πΉπ‘›+1ξƒ©π‘βˆšπ‘žξƒͺ,π‘Žπ‘›ξƒ©ξ‚™=π‘βˆ’π‘ž2ξƒͺ𝑛Φ𝑛+1𝑝2ξ‚Ά,π‘Žβˆ’π‘žπ‘›=𝑐2𝑛+2ξƒ©ξ‚™βˆ’π‘žξƒͺ4π‘Žπ‘›π·π‘›ξ‚΅π‘ξ‚™π‘Žξ‚Ά,π‘Žβˆ’π‘ž,π‘Žπ‘›ξ‚€βˆ’βˆš=π‘ξ‚βˆ’π‘žπ‘›π‘ˆπ‘›ξƒ©βˆ’π‘2√ξƒͺ,π‘Žβˆ’π‘žπ‘›ξ‚€βˆ’βˆš=π‘π‘žξ‚π‘›π‘ƒπ‘›+1ξƒ©βˆ’π‘2βˆšπ‘žξƒͺ,π‘Žπ‘›ξ‚€βˆ’βˆš=π‘π‘žξ‚π‘›πΉπ‘›+1ξƒ©βˆ’π‘βˆšπ‘žξƒͺ,π‘Žπ‘›ξƒ©βˆ’ξ‚™=π‘βˆ’π‘ž2ξƒͺ𝑛Φ𝑛+1ξ‚΅ξ‚™βˆ’π‘2ξ‚Ά,π‘Žβˆ’π‘žπ‘›=𝑐2𝑛+2ξƒ©βˆ’ξ‚™βˆ’π‘žξƒͺ4π‘Žπ‘›π·π‘›ξ‚΅ξ‚™βˆ’π‘π‘Žξ‚Ά.βˆ’π‘ž,π‘Ž(2.11)

If π‘Ž1=𝑑=1, then Corollary 2.2 gives the primary solutions of recurrence relation (1.1) in terms of the 𝑛th degree Chebyshev polynomial of the second kind, the Pell polynomial, the Fibonacci polynomial, the Fermat polynomial, and the Dickson polynomial of the second kind, respectively. For instance, if 𝑝=π‘ž=1, then π‘Žπ‘› are the Fibonacci numbers 𝐹𝑛. Thus, 𝐹𝑛=(𝑖)π‘›βˆ’1π‘ˆπ‘›βˆ’1ξ‚€12𝑖=(𝑖)π‘›βˆ’1π‘ˆπ‘›βˆ’1ξ‚€βˆ’π‘–2,𝐹𝑛=𝑃𝑛12,𝐹𝑛=𝐹𝑛𝐹(1),𝑛=ξƒ©π‘–βˆš2ξƒͺπ‘›βˆ’1Ξ¦π‘›ξ‚€βˆ’βˆšξ‚,𝐹2𝑖𝑛=2𝑛+1ξƒ©π‘–βˆšξƒͺ4π‘Žπ‘›βˆ’1π·π‘›βˆ’1ξ‚€βˆ’βˆšξ‚,πΉπ‘Žπ‘–,π‘Žπ‘›=(βˆ’π‘–)π‘›βˆ’1π‘ˆπ‘›βˆ’1𝑖2,𝐹𝑛=(βˆ’1)π‘›βˆ’1π‘ƒπ‘›ξ‚€βˆ’12,𝐹𝑛=(βˆ’1)π‘›βˆ’1𝐹𝑛𝐹(βˆ’1),𝑛=ξƒ©βˆ’π‘–βˆš2ξƒͺπ‘›βˆ’1Ξ¦π‘›ξ‚€βˆšξ‚,𝐹2𝑖𝑛=2𝑛+1ξƒ©βˆ’π‘–βˆšξƒͺ4π‘Žπ‘›βˆ’1π·π‘›βˆ’1ξ‚€βˆšξ‚,π‘Žπ‘–,π‘Ž(2.12) where 𝐹𝑛=(𝑖)π‘›βˆ’1π‘ˆπ‘›βˆ’1(βˆ’π‘–/2) was shown in [8] and 𝐹𝑛=(βˆ’π‘–)π‘›βˆ’1π‘ˆπ‘›βˆ’1(𝑖/2) was given by Chen and Louck in [17]. From the above expressions of 𝐹𝑛, we may obtain many identities. For instance, we have 𝑃𝑛12=(βˆ’1)π‘›βˆ’1π‘ƒπ‘›ξ‚€βˆ’12=𝐹𝑛(1)=(βˆ’1)π‘›βˆ’1𝐹𝑛(βˆ’1),(𝑖)π‘›βˆ’1π‘ˆπ‘›βˆ’1ξ‚€βˆ’π‘–2=(βˆ’π‘–)π‘›βˆ’1π‘ˆπ‘›βˆ’1𝑖2=ξƒ©π‘–βˆš2ξƒͺπ‘›βˆ’1Ξ¦π‘›ξ‚€βˆ’βˆšξ‚=ξƒ©βˆ’π‘–2π‘–βˆš2ξƒͺπ‘›βˆ’1Ξ¦π‘›ξ‚€βˆšξ‚,2𝑖(2.13) and so forth.

We now give another special case of Theorem 2.1 for the sequence defined by (1.1) with initial cases π‘Ž0=2 and π‘Ž1.

Corollary 2.4. Let sequence {π‘Žπ‘›} be defined by π‘Žπ‘›=π‘π‘Žπ‘›βˆ’1+π‘žπ‘Žπ‘›βˆ’2 (𝑛β‰₯2) with initial conditions π‘Ž0=2 and π‘Ž1=𝑝. Then π‘Žπ‘›ξ‚€βˆš=2ξ‚βˆ’π‘žπ‘›π‘ˆπ‘›ξƒ©π‘2√ξƒͺξ‚€βˆšβˆ’π‘žβˆ’π‘ξ‚βˆ’π‘žπ‘›βˆ’1π‘ˆπ‘›βˆ’1𝑝2√ξƒͺ,π‘Žβˆ’π‘žπ‘›ξ‚€βˆš=2π‘žξ‚π‘›π‘ƒπ‘›+1𝑝2βˆšπ‘žξƒͺξ‚€βˆšβˆ’π‘π‘žξ‚π‘›βˆ’1𝑃𝑛𝑝2βˆšπ‘žξƒͺ,π‘Žπ‘›ξ‚€βˆš=2π‘žξ‚π‘›πΉπ‘›+1ξƒ©π‘βˆšπ‘žξƒͺξ‚€βˆšβˆ’π‘π‘žξ‚π‘›βˆ’1πΉπ‘›ξƒ©π‘βˆšπ‘žξƒͺ,π‘Žπ‘›ξƒ©ξ‚™=2βˆ’π‘ž2ξƒͺ𝑛Φ𝑛+1𝑝2ξ‚Άξƒ©ξ‚™βˆ’π‘žβˆ’π‘βˆ’π‘ž2ξƒͺπ‘›βˆ’1Φ𝑛𝑝2ξ‚Ά,π‘Žβˆ’π‘žπ‘›=2𝑛+3ξƒ©ξ‚™βˆ’π‘žξƒͺ4π‘Žπ‘›π·π‘›ξ‚΅π‘ξ‚™π‘Žξ‚Άβˆ’π‘ž,π‘Žβˆ’π‘2𝑛+1ξƒ©ξ‚™βˆ’π‘žξƒͺ4π‘Žπ‘›βˆ’1π·π‘›βˆ’1ξ‚΅π‘ξ‚™π‘Žξ‚Ά,π‘Žβˆ’π‘ž,π‘Žπ‘›ξ‚€βˆ’βˆš=2ξ‚βˆ’π‘žπ‘›π‘ˆπ‘›ξƒ©βˆ’π‘2√ξƒͺξ‚€βˆ’βˆšβˆ’π‘žβˆ’π‘ξ‚βˆ’π‘žπ‘›βˆ’1π‘ˆπ‘›βˆ’1ξƒ©βˆ’π‘2√ξƒͺ,π‘Žβˆ’π‘žπ‘›ξ‚€βˆ’βˆš=2π‘žξ‚π‘›π‘ƒπ‘›+1ξƒ©βˆ’π‘2βˆšπ‘žξƒͺξ‚€βˆ’βˆšβˆ’π‘π‘žξ‚π‘›βˆ’1π‘ƒπ‘›ξƒ©βˆ’π‘2βˆšπ‘žξƒͺ,π‘Žπ‘›ξ‚€βˆ’βˆš=2π‘žξ‚π‘›πΉπ‘›+1ξƒ©βˆ’π‘βˆšπ‘žξƒͺξ‚€βˆ’βˆšβˆ’π‘π‘žξ‚π‘›βˆ’1πΉπ‘›ξƒ©βˆ’π‘βˆšπ‘žξƒͺ,π‘Žπ‘›ξƒ©βˆ’ξ‚™=2βˆ’π‘ž2ξƒͺ𝑛Φ𝑛+1ξ‚΅ξ‚™βˆ’π‘2ξ‚Άξƒ©βˆ’ξ‚™βˆ’π‘žβˆ’π‘βˆ’π‘ž2ξƒͺπ‘›βˆ’1Ξ¦π‘›ξ‚΅ξ‚™βˆ’π‘2ξ‚Ά,π‘Žβˆ’π‘žπ‘›=2𝑛+3ξƒ©βˆ’ξ‚™βˆ’π‘žξƒͺ4π‘Žπ‘›π·π‘›ξ‚΅ξ‚™βˆ’π‘π‘Žξ‚Άβˆ’π‘ž,π‘Žβˆ’π‘2𝑛+1ξƒ©βˆ’ξ‚™βˆ’π‘žξƒͺ4π‘Žπ‘›βˆ’1π·π‘›βˆ’1ξ‚΅ξ‚™βˆ’π‘π‘Žξ‚Ά.βˆ’π‘ž,π‘Ž(2.14) In addition, one has π‘Žπ‘›ξ‚€βˆš=2ξ‚βˆ’π‘žπ‘›π‘‡π‘›ξƒ©π‘2√ξƒͺπ‘Žβˆ’π‘ž,(2.15)π‘›ξ‚€βˆ’βˆš=2ξ‚βˆ’π‘žπ‘›π‘‡π‘›ξƒ©βˆ’π‘2√ξƒͺβˆ’π‘ž,(2.16) where 𝑇𝑛(π‘₯) are the Chebyshev polynomials of the first kind.

Proof. It is sufficient to prove (2.15) and (2.16). From the first formula shown in Corollary 2.4 and the recurrence relation π‘ˆπ‘›(π‘₯)=2π‘₯π‘ˆπ‘›βˆ’1(π‘₯)βˆ’π‘ˆπ‘›βˆ’2(π‘₯), one easily sees π‘Žπ‘›=ξ‚€βˆšξ‚βˆ’π‘žπ‘›ξƒ¬2π‘ˆπ‘›ξƒ©π‘2√ξƒͺβˆ’π‘βˆ’π‘žβˆšπ‘ˆβˆ’π‘žπ‘›βˆ’1𝑝2√=ξ‚€βˆšβˆ’π‘žξƒͺξƒ­ξ‚βˆ’π‘žπ‘›ξƒ¬2π‘ˆπ‘›ξƒ©π‘2√ξƒͺβˆ’ξƒ©π‘ˆβˆ’π‘žπ‘›ξƒ©π‘2√ξƒͺβˆ’π‘ž+π‘ˆπ‘›βˆ’2𝑝2√=ξ‚€βˆšβˆ’π‘žξƒͺξƒͺξƒ­ξ‚βˆ’π‘žπ‘›ξƒ¬π‘ˆπ‘›ξƒ©π‘2√ξƒͺβˆ’π‘žβˆ’π‘ˆπ‘›βˆ’2𝑝2√.βˆ’π‘žξƒͺξƒ­(2.17) From the basic relation between Chebyshev polynomials of the first and the second kinds (see, e.g., (1.7) in [18] by Mason and Handscomb), π‘ˆπ‘›(π‘₯)βˆ’π‘ˆπ‘›βˆ’2(π‘₯)=2𝑇𝑛(π‘₯), the last expression of π‘Žπ‘› implies (2.15). Equation (2.16) can be proved similarly.

As an example, the Lucas number sequence {𝐿𝑛} defined by (1.1) with 𝑝=π‘ž=1 and initial conditions 𝐿0=2 and 𝐿1=1 has the explicit formula for its general term: 𝐿𝑛=2π‘–π‘›π‘‡π‘›ξ‚€βˆ’π‘–2=2(βˆ’π‘–)𝑛𝑇𝑛𝑖2.(2.18)

3. Examples and Applications

We first give some examples of Corollary 2.2 for sequences {π‘Žπ‘›} that are primary solutions of (1.1).

Example 3.1. If 𝑝=2 and π‘ž=1, then π‘Žπ‘› defined by (1.1) with initial conditions π‘Ž0=0 and π‘Ž1=1 are the Pell numbers 𝑃𝑛. Thus, from Corollary 2.2, we have 𝑃𝑛=(𝑖)π‘›βˆ’1π‘ˆπ‘›βˆ’1(βˆ’π‘–)=(βˆ’π‘–)π‘›βˆ’1π‘ˆπ‘›βˆ’1𝑃(𝑖),𝑛=𝑃𝑛(1)=(βˆ’1)π‘›βˆ’1𝑃𝑛𝑃(βˆ’1),𝑛=𝐹𝑛(2)=(βˆ’1)π‘›βˆ’1𝐹𝑛𝑃(βˆ’2),𝑛=ξƒ©π‘–βˆš2ξƒͺπ‘›βˆ’1Ξ¦π‘›ξ‚€βˆšβˆ’2=ξƒ©βˆ’π‘–2π‘–βˆš2ξƒͺπ‘›βˆ’1Φ𝑛2βˆšξ‚,𝑃2𝑖𝑛=2𝑛+1ξƒ©π‘–βˆšξƒͺ4π‘Žπ‘›βˆ’1π·π‘›βˆ’1ξ‚€βˆšβˆ’2ξ‚π‘Žπ‘–,π‘Ž=2𝑛+1ξƒ©βˆ’π‘–βˆšξƒͺ4π‘Žπ‘›βˆ’1π·π‘›βˆ’1ξ‚€2βˆšξ‚.π‘Žπ‘–,π‘Ž(3.1)

Example 3.2. If 𝑝=1 and π‘ž=2, then π‘Žπ‘› defined by (1.1) with initial conditions π‘Ž0=0 and π‘Ž1=1 are the Jacobsthal numbers 𝐽𝑛 (see Bergum et al. [19]). Thus Corollary 2.2 gives the expressions of 𝐽𝑛 as follows: 𝐽𝑛=ξ‚€βˆšξ‚2π‘–π‘›βˆ’1π‘ˆπ‘›βˆ’1ξƒ©βˆ’π‘–2√2ξƒͺ=ξ‚€βˆ’βˆšξ‚2π‘–π‘›βˆ’1π‘ˆπ‘›βˆ’1𝑖2√2ξƒͺ,𝐽𝑛=ξ‚€βˆš2ξ‚π‘›βˆ’1𝑃𝑛12√2ξƒͺ=ξ‚€βˆ’βˆš2ξ‚π‘›βˆ’1π‘ƒπ‘›ξƒ©βˆ’12√2ξƒͺ,𝐽𝑛=ξ‚€βˆš2ξ‚π‘›βˆ’1𝐹𝑛1√2ξƒͺ=ξ‚€βˆ’βˆš2ξ‚π‘›βˆ’1πΉπ‘›ξƒ©βˆ’1√2ξƒͺ,𝐽𝑛=π‘–π‘›βˆ’1Φ𝑛(βˆ’π‘i)=(βˆ’π‘–)π‘›βˆ’1Φ𝑛𝐽(𝑝𝑖),𝑛=2𝑛+1ξƒ©π‘–βˆšξƒͺ2π‘Žπ‘›βˆ’1π·π‘›βˆ’1ξƒ©βˆ’π‘βˆšπ‘Žπ‘–βˆš2ξƒͺ,π‘Ž=2𝑛+1ξƒ©βˆ’π‘–βˆšξƒͺ2π‘Žπ‘›βˆ’1π·π‘›βˆ’1ξƒ©π‘βˆšπ‘Žπ‘–βˆš2ξƒͺ.,π‘Ž(3.2)

Example 3.3. If 𝑝=3 and π‘ž=βˆ’2, then π‘Žπ‘› defined by (1.1) with initial conditions π‘Ž0=0 and π‘Ž1=1 are the Mersenne numbers 𝑀𝑛=2π‘›βˆ’1. From Corollary 2.2, we have 𝑀𝑛=ξ‚€βˆš2ξ‚π‘›βˆ’1π‘ˆπ‘›βˆ’132√2ξƒͺ=ξ‚€βˆ’βˆš2ξ‚π‘›βˆ’1π‘ˆπ‘›βˆ’1ξƒ©βˆ’32√2ξƒͺ,𝑀𝑛=ξ‚€βˆšξ‚2π‘–π‘›βˆ’1π‘ƒπ‘›ξƒ©βˆ’3𝑖2√2ξƒͺ=ξ‚€βˆ’βˆšξ‚2π‘–π‘›βˆ’1𝑃𝑛3𝑖2√2ξƒͺ,𝑀𝑛=ξ‚€βˆšξ‚2π‘–π‘›βˆ’1πΉπ‘›ξƒ©βˆ’3π‘–βˆš2ξƒͺ=ξ‚€βˆ’βˆšξ‚2π‘–π‘›βˆ’1𝐹𝑛3π‘–βˆš2ξƒͺ,𝑀𝑛=Φ𝑛(3)=(βˆ’1)π‘›βˆ’1Φ𝑛𝑀(βˆ’3),𝑛=2𝑛+11√ξƒͺ2π‘Žπ‘›βˆ’1π·π‘›βˆ’13βˆšπ‘Žβˆš2ξƒͺ,π‘Ž=2𝑛+1ξƒ©βˆ’1√ξƒͺ2π‘Žπ‘›βˆ’1π·π‘›βˆ’1ξƒ©βˆ’3βˆšπ‘Žβˆš2ξƒͺ.,π‘Ž(3.3)

Next, we give several examples of nonprimary solutions of (1.1) by using Corollary 2.4.

Example 3.4. If 𝑝=1 and π‘ž=1, then π‘Žπ‘› defined by (1.1) with initial conditions π‘Ž0=2 and π‘Ž1=1 are the Lucas numbers 𝐿𝑛. Thus, besides (2.18), we have 𝐿𝑛=2π‘–π‘›π‘ˆπ‘›ξ‚€βˆ’π‘–2ξ‚βˆ’π‘–π‘›βˆ’1π‘ˆπ‘›βˆ’1ξ‚€βˆ’π‘–2=2(βˆ’π‘–)π‘›π‘ˆπ‘›ξ‚€π‘–2ξ‚βˆ’(βˆ’π‘–)π‘›βˆ’1π‘ˆπ‘›βˆ’1𝑖2,𝐿𝑛=2𝑃𝑛+1ξ‚€12ξ‚βˆ’π‘ƒπ‘›ξ‚€12=2(βˆ’1)𝑛𝑃𝑛+1ξ‚€βˆ’12ξ‚βˆ’(βˆ’1)π‘›βˆ’1π‘ƒπ‘›ξ‚€βˆ’12,𝐿𝑛=2𝐹𝑛+1(1)βˆ’πΉπ‘›(1)=2(βˆ’1)𝑛𝐹𝑛+1(βˆ’1)βˆ’(βˆ’1)π‘›βˆ’1𝐹𝑛𝐿(βˆ’1),𝑛𝑖=2√2ξƒͺ𝑛Φ𝑛+1ξ‚€βˆ’βˆšξ‚βˆ’ξƒ©π‘–2π‘–βˆš2ξƒͺπ‘›βˆ’1Ξ¦π‘›ξ‚€βˆ’βˆšξ‚ξƒ©βˆ’π‘–2𝑖=2√2ξƒͺ𝑛Φ𝑛+1ξ‚€βˆšξ‚βˆ’ξƒ©βˆ’π‘–2π‘–βˆš2ξƒͺπ‘›βˆ’1Ξ¦π‘›ξ‚€βˆšξ‚,𝐿2𝑖𝑛=2𝑛+3ξƒ©π‘–βˆšξƒͺ4π‘Žπ‘›π·π‘›ξ‚€βˆ’βˆšξ‚π‘Žπ‘–,π‘Žβˆ’2𝑛+1ξƒ©π‘–βˆšξƒͺ4π‘Žπ‘›βˆ’1π·π‘›βˆ’1ξ‚€βˆ’βˆšξ‚π‘Žπ‘–,π‘Ž=2𝑛+3ξƒ©βˆ’π‘–βˆšξƒͺ4π‘Žπ‘›π·π‘›ξ‚€βˆšξ‚π‘Žπ‘–,π‘Žβˆ’2𝑛+1ξƒ©βˆ’π‘–βˆšξƒͺ4π‘Žπ‘›βˆ’1π·π‘›βˆ’1ξ‚€βˆšξ‚.π‘Žπ‘–,π‘Ž(3.4)

Example 3.5. If 𝑝=2 and π‘ž=1, then π‘Žπ‘› defined by (1.1) with initial conditions π‘Ž0=2 and π‘Ž1=2 are the Pell-Lucas numbers 𝐴𝑛 (see Example 2 in [11]). Thus, from Corollary 2.4, we obtain 𝐴𝑛=2𝑖𝑛𝑇𝑛(βˆ’π‘–)=2(βˆ’π‘–)𝑛𝑇𝑛𝐴(𝑖),𝑛=2π‘–π‘›π‘ˆπ‘›(βˆ’π‘–)βˆ’2π‘–π‘›βˆ’1π‘ˆπ‘›βˆ’1(βˆ’π‘–)=2π‘–π‘›π‘ˆπ‘›(βˆ’π‘–)βˆ’2π‘–π‘›βˆ’1π‘ˆπ‘›βˆ’1(π΄βˆ’π‘–),𝑛=2𝑃𝑛+1(1)βˆ’2𝑃𝑛(1)=2(βˆ’1)𝑛𝑃𝑛+1(βˆ’1)βˆ’π‘(βˆ’1)π‘›βˆ’1𝑃𝑛(π΄βˆ’1),𝑛=2𝐹𝑛+1(2)βˆ’2𝐹𝑛(2)=2(βˆ’1)𝑛𝐹𝑛+1(βˆ’2)βˆ’π‘(βˆ’1)π‘›βˆ’1𝐹𝑛(π΄βˆ’2),π‘›ξƒ©βˆ’π‘–=2√2ξƒͺ𝑛Φ𝑛+1ξ‚€2βˆšξ‚ξƒ©βˆ’π‘–2π‘–βˆ’2√2ξƒͺπ‘›βˆ’1Φ𝑛2βˆšξ‚ξƒ©π‘–2𝑖=2√2ξƒͺ𝑛Φ𝑛+1ξ‚€βˆšβˆ’2𝑖2π‘–βˆ’2√2ξƒͺπ‘›βˆ’1Ξ¦π‘›ξ‚€βˆšβˆ’2,𝐴2𝑖𝑛=2𝑛+3ξƒ©π‘–βˆšξƒͺ4π‘Žπ‘›π·π‘›ξ‚€βˆšβˆ’2ξ‚π‘Žπ‘–,π‘Žβˆ’2𝑛+2ξƒ©π‘–βˆšξƒͺ4π‘Žπ‘›βˆ’1π·π‘›βˆ’1ξ‚€βˆšβˆ’2ξ‚π‘Žπ‘–,π‘Ž=2𝑛+3ξƒ©βˆ’π‘–βˆšξƒͺ4π‘Žπ‘›π·π‘›ξ‚€2βˆšξ‚π‘Žπ‘–,π‘Žβˆ’2𝑛+2ξƒ©βˆ’π‘–βˆšξƒͺ4π‘Žπ‘›βˆ’1π·π‘›βˆ’1ξ‚€2βˆšξ‚.π‘Žπ‘–,π‘Ž(3.5)

Example 3.6. If 𝑝=1 and π‘ž=2, then π‘Žπ‘› defined by (1.1) with initial conditions π‘Ž0=2 and π‘Ž1=1 are the Jacobsthal-Lucas numbers 𝐡𝑛 (see Example 2 in [11]). Thus, π΅π‘›ξ‚€βˆš=22π‘–π‘›π‘‡π‘›ξƒ©βˆ’π‘–2√2ξƒͺξ‚€βˆ’βˆš=22𝑖𝑛𝑇𝑛𝑖2√2ξƒͺ,π΅π‘›ξ‚€βˆš=22π‘–π‘›π‘ˆπ‘›ξƒ©βˆ’π‘–2√2ξƒͺβˆ’ξ‚€βˆšξ‚2π‘–π‘›βˆ’1π‘ˆπ‘›βˆ’1ξƒ©βˆ’π‘–2√2ξƒͺξ‚€βˆ’βˆš=22π‘–π‘›π‘ˆπ‘›ξƒ©π‘–2√2ξƒͺβˆ’ξ‚€βˆ’βˆšξ‚2π‘–π‘›βˆ’1π‘ˆπ‘›βˆ’1𝑖2√2ξƒͺ,π΅π‘›ξ‚€βˆš=22𝑛𝑃𝑛+112√2ξƒͺβˆ’ξ‚€βˆš2ξ‚π‘›βˆ’1𝑃𝑛12√2ξƒͺξ‚€βˆ’βˆš=22𝑛𝑃𝑛+1ξƒ©βˆ’12√2ξƒͺβˆ’ξ‚€βˆ’βˆš2ξ‚π‘›βˆ’1π‘ƒπ‘›ξƒ©βˆ’12√2ξƒͺ,π΅π‘›ξ‚€βˆš=22𝑛𝐹𝑛+11√2ξƒͺβˆ’ξ‚€βˆš2ξ‚π‘›βˆ’1𝐹𝑛1√2ξƒͺξ‚€βˆ’βˆš=22𝑛𝐹𝑛+1ξƒ©βˆ’1√2ξƒͺβˆ’ξ‚€βˆ’βˆš2ξ‚π‘›βˆ’1πΉπ‘›ξƒ©βˆ’1√2ξƒͺ,𝐡𝑛=2𝑖𝑛Φ𝑛+1(βˆ’π‘–)βˆ’π‘–π‘›βˆ’1Φ𝑛(βˆ’π‘–)=2(βˆ’π‘–)𝑛Φ𝑛+1(𝑖)βˆ’(βˆ’π‘–)π‘›βˆ’1Φ𝑛𝐡(𝑖),𝑛=2𝑛+3ξƒ©π‘–βˆšξƒͺ2π‘Žπ‘›π·π‘›ξƒ©βˆ’βˆšπ‘Žπ‘–βˆš2ξƒͺ,π‘Žβˆ’2𝑛+1ξƒ©π‘–βˆšξƒͺ2π‘Žπ‘›βˆ’1π·π‘›βˆ’1ξƒ©βˆ’βˆšπ‘Žπ‘–βˆš2ξƒͺ,π‘Ž=2𝑛+3ξƒ©βˆ’π‘–βˆšξƒͺ2π‘Žπ‘›π·π‘›ξƒ©βˆšπ‘Žπ‘–βˆš2ξƒͺ,π‘Žβˆ’2𝑛+1ξƒ©βˆ’π‘–βˆšξƒͺ2π‘Žπ‘›βˆ’1π·π‘›βˆ’1ξƒ©βˆšπ‘Žπ‘–βˆš2ξƒͺ.,π‘Ž(3.6)

Example 3.7. If 𝑝=3 and π‘ž=βˆ’2, then π‘Žπ‘› defined by (1.1) with initial conditions π‘Ž0=2 and π‘Ž1=3 are the Fermat numbers 𝑓𝑛 (see [20]). Thus, from Corollary 2.4, we obtain π‘“π‘›ξ‚€βˆš=22𝑛𝑇𝑛32√2ξƒͺξ‚€βˆ’βˆš=22ξ‚π‘›π‘‡π‘›ξƒ©βˆ’32√2ξƒͺ,π‘“π‘›ξ‚€βˆš=22ξ‚π‘›π‘ˆπ‘›ξƒ©32√2ξƒͺξ‚€βˆšβˆ’32ξ‚π‘›βˆ’1π‘ˆπ‘›βˆ’132√2ξƒͺξ‚€βˆ’βˆš=22ξ‚π‘›π‘ˆπ‘›ξƒ©βˆ’32√2ξƒͺξ‚€βˆ’βˆšβˆ’32ξ‚π‘›βˆ’1π‘ˆπ‘›βˆ’1ξƒ©βˆ’32√2ξƒͺ,π‘“π‘›ξ‚€βˆš=22𝑖𝑛𝑃𝑛+1ξƒ©βˆ’3𝑖2√2ξƒͺξ‚€βˆšβˆ’32π‘–π‘›βˆ’1π‘ƒπ‘›ξƒ©βˆ’3𝑖2√2ξƒͺξ‚€βˆ’βˆš=22𝑖𝑛𝑃𝑛+13𝑖2√2ξƒͺξ‚€βˆ’βˆšβˆ’32π‘–π‘›βˆ’1𝑃𝑛3𝑖2√2ξƒͺ,π‘“π‘›ξ‚€βˆš=22𝑖𝑛𝐹𝑛+1ξƒ©βˆ’3π‘–βˆš2ξƒͺξ‚€βˆšβˆ’32π‘–π‘›βˆ’1πΉπ‘›ξƒ©βˆ’3π‘–βˆš2ξƒͺξ‚€βˆ’βˆš=22𝑖𝑛𝐹𝑛+13π‘–βˆš2ξƒͺξ‚€βˆ’βˆšβˆ’32π‘–π‘›βˆ’1𝐹𝑛3π‘–βˆš2ξƒͺ,𝑓𝑛=2Φ𝑛+1(3)βˆ’3Φ𝑛(3)=2(βˆ’1)𝑛Φ𝑛+1(βˆ’3)βˆ’3(βˆ’1)π‘›βˆ’1Φ𝑛𝑓(βˆ’3),𝑛=2𝑛+31√ξƒͺ2π‘Žπ‘›π·π‘›ξƒ©3βˆšπ‘Žβˆš2ξƒͺ,π‘Žβˆ’(3)2𝑛+11√ξƒͺ2π‘Žπ‘›βˆ’1π·π‘›βˆ’13βˆšπ‘Žβˆš2ξƒͺ,π‘Ž=2𝑛+3ξƒ©βˆ’1√ξƒͺ2π‘Žπ‘›π·π‘›ξƒ©βˆ’3βˆšπ‘Žβˆš2ξƒͺβˆ’,π‘Ž(3)2𝑛+1ξƒ©βˆ’1√ξƒͺ2π‘Žπ‘›βˆ’1π·π‘›βˆ’1ξƒ©βˆ’3βˆšπ‘Žβˆš2ξƒͺ.,π‘Ž(3.7)

Using the relationship established above, we may obtain some identities of number sequences and polynomial value sequences. Theorem 3.2 in [11] presented a generalized Gegenbauer-Humbert polynomial sequence identity: 𝑃𝑛1,𝑦,𝐢(π‘₯)=𝛼(π‘₯)𝑃1,𝑦,πΆπ‘›βˆ’1(π‘₯)+πΆβˆ’2(2π‘₯βˆ’π›Ό(π‘₯)𝐢)(𝛽(π‘₯))π‘›βˆ’1,(3.8) where 𝑃𝑛1,𝑦,𝐢(π‘₯) satisfies the recurrence relation of order 2, 𝑃𝑛1,𝑦,𝐢=𝑝𝑃1,𝑦,πΆπ‘›βˆ’1+π‘žπ‘ƒ1,𝑦,πΆπ‘›βˆ’2 with coefficients 𝑝(π‘₯) and π‘ž(π‘₯), and 𝛼(π‘₯)+𝛽(π‘₯)=𝑝(π‘₯) and 𝛼(π‘₯)𝛽(π‘₯)=βˆ’π‘ž(π‘₯). Clearly (see (19) and (20) in [11]), 1𝛼=πΆξ‚†βˆšπ‘₯+π‘₯2,1βˆ’πΆπ‘¦π›½=πΆξ‚†βˆšπ‘₯βˆ’π‘₯2.βˆ’πΆπ‘¦(3.9) For 𝑦=βˆ’1 and 𝐢=1, we have 𝑃𝑛1,βˆ’1,1(π‘₯)=𝐹𝑛+1(2π‘₯), where 𝐹𝑛(π‘₯) are the Fibonacci polynomials, and we can write (3.8) as 𝐹𝑛+1(2π‘₯)=𝛼(π‘₯)𝐹𝑛(2π‘₯)+(2π‘₯βˆ’π›Ό(π‘₯))(𝛽(π‘₯))π‘›βˆ’1=𝛼(π‘₯)𝐹𝑛(2π‘₯)+(𝛽(π‘₯))𝑛,(3.10) where βˆšπ›Ό(π‘₯)=π‘₯+π‘₯2+1 and βˆšπ›½(π‘₯)=π‘₯βˆ’π‘₯2+1. If π‘₯=1/2, then 𝐹𝑛(1)=𝐹𝑛, the Fibonacci numbers, and 𝛼12=√1+52ξ‚€1,𝛽2=√1βˆ’52.(3.11)

Thus (3.10) yields the identity 𝐹𝑛+1=√1+52𝐹𝑛+ξƒ©βˆš1βˆ’52ξƒͺ𝑛,(3.12) or equivalently, √1βˆ’52𝐹𝑛+1+𝐹𝑛=ξƒ©βˆš1βˆ’52ξƒͺ𝑛+1.(3.13)

Similarly, if π‘₯=1, then 𝐹𝑛(2)=𝑃𝑛, the Pell numbers, and βˆšπ›Ό(1)=1+√2,𝛽(1)=1βˆ’2.(3.14) Thus (3.10) yields the identity𝑃𝑛+1=ξ‚€βˆš1+2𝑃𝑛+ξ‚€βˆš1βˆ’2𝑛,(3.15) or equivalently, ξ‚€βˆš1βˆ’2𝑃𝑛+1+𝑃𝑛=ξ‚€βˆš1βˆ’2𝑛+1.(3.16)

Substituting √π‘₯=1/(22) into (3.10) and noting πΉπ‘›βˆš(1/2)=π½π‘›βˆš/(2)𝑛, where 𝐽𝑛 are the Jacobsthal numbers, we obtain the identity 𝐽𝑛+1βˆ’2𝐽𝑛=(βˆ’1)𝑛.(3.17) When √π‘₯=βˆ’3𝑖/(22), πΉπ‘›βˆš(βˆ’3𝑖/(22))=π‘€π‘›βˆš/(2𝑖)π‘›βˆ’1, the Mersenne numbers. Hence (3.10) gives 𝑀𝑛+1βˆ’π‘€π‘›=2𝑛.

Conversely, one may use the expressions of various number sequences in terms of the generalized Gegenbauer-Humbert polynomial sequences to construct the identities of the different generalized Gegenbauer-Humbert polynomial values such as the formulas shown in the example after Corollary 2.3.

Acknowledgments

P. J.-S. Shiue and T.-W. Weng would like to thank the Institute of Mathematics, Academia Sinica, Taiwan, for its financial support of the research in this paper carried out during summer 2009.

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