Sequences of Numbers Meet the Generalized Gegenbauer-Humbert Polynomials

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 {𝑎_𝑛} is called sequence of order 2 if it satisfies the linear recurrence relation of order 2: 𝑎_𝑛=𝑝𝑎_𝑛−1+𝑞𝑎_𝑛−2,𝑛≥2,(1.1) for some nonzero constants 𝑝 and 𝑞 and initial conditions 𝑎₀ and 𝑎₁. In Mansour [1], the sequence {𝑎_𝑛}_𝑛≥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, 𝑎_𝑛 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 𝑞 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 𝑎₀=0 and 𝑎₁=1, called the primary solution, can be expressed in terms of the Chebyshev polynomial values. For instance, the authors show 𝐹_𝑛=𝑖^−𝑛𝑈_𝑛(𝑖/2) and 𝐿_𝑛=2𝑖^−𝑛𝑇_𝑛(𝑖/2), where 𝐹_𝑛 and 𝐿_𝑛, respectively, are the Fibonacci numbers and Lucas numbers, and 𝑇_𝑛 and 𝑈_𝑛 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 {𝑎_𝑛} generated by (1.1). For the sake of the reader's convenience, we cite this result as follows.

Proposition 1.1 (see [11]). Let {𝑎_𝑛} be a sequence of order 2 satisfying linear recurrence relation (1.1), and let 𝛼 and 𝛽 be two roots of of quadratic equation 𝑥²−𝑝𝑥−𝑞=0. Then 𝑎_𝑛=⎧⎪⎨⎪⎩𝑎¹−𝛽𝑎₀𝛼−𝛽𝛼^𝑛−𝑎¹−𝛼𝑎₀𝛼−𝛽𝛽^𝑛𝑛𝑎₁,if𝛼≠𝛽,𝛼^𝑛−1−(𝑛−1)𝑎₀𝛼^𝑛,if𝛼=𝛽.(1.2)

A sequence of the generalized Gegenbauer-Humbert polynomials {𝑃^{𝑛𝜆,𝑦,𝐶}(𝑥)}_𝑛≥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]) Φ(𝑡)≡𝐶−2𝑥𝑡+𝑦𝑡²^−𝜆=_𝑛≥0𝑃^{𝑛𝜆,𝑦,𝐶}(𝑥)𝑡^𝑛,(1.3) where 𝜆>0, 𝑦 and 𝐶≠0 are real numbers. As special cases of (1.3), we consider 𝑃^{𝑛𝜆,𝑦,𝐶}(𝑥) as follows (see [14]): 𝑃^𝑛1,1,1(𝑥)=𝑈_𝑛(𝑥), the Chebyshev polynomial of the second kind,𝑃^𝑛1/2,1,1(𝑥)=𝜓_𝑛(𝑥), the Legendre polynomial,𝑃^𝑛1,−1,1(𝑥)=𝑃_𝑛+1(𝑥), the Pell polynomial,𝑃^𝑛1,−1,1(𝑥/2)=𝐹_𝑛+1(𝑥), the Fibonacci polynomial,𝑃^𝑛1,2,1(𝑥/2)=Φ_𝑛+1(𝑥), the Fermat polynomial of the first kind,𝑃^{𝑛1,2𝑎,2}(𝑥)=𝐷_𝑛(𝑥,𝑎), the Dickson polynomial of the second kind, 𝑎≠0 (see, e.g., [13]),

where 𝑎 is a real parameter, and 𝐹_𝑛=𝐹_𝑛(1) is the Fibonacci number. In particular, if 𝑦=𝐶=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 𝑃^{𝑛𝜆,𝑦,𝐶}(𝑥)=2𝑥𝜆+𝑛−1𝐶𝑛𝑃^{𝜆,𝑦,𝐶𝑛−1}(𝑥)−𝑦2𝜆+𝑛−2𝐶𝑛𝑃^{𝜆,𝑦,𝐶𝑛−2}(𝑥),(1.4) for all 𝑛≥2 with initial conditions𝑃𝑃^{0𝜆,𝑦,𝐶}_{1𝜆,𝑦,𝐶}(𝑥)=Φ(0)=𝐶^−𝜆,(𝑥)=Φ^(0)=2𝜆𝑥𝐶^−𝜆−1,(1.5) the following theorem has been obtained in [11].

Theorem 1.2 (see [11]). Let 𝑥≠±√𝐶𝑦. The generalized Gegenbauer-Humbert polynomials {𝑃^{𝑛1,𝑦,𝐶}(𝑥)}_𝑛≥0 defined by expansion (1.3) can be expressed as 𝑃^{𝑛1,𝑦,𝐶}(𝑥)=𝐶^−𝑛−2𝑥+√𝑥²−𝐶𝑦^𝑛+1−𝑥−√𝑥²−𝐶𝑦^𝑛+12√𝑥²−𝐶𝑦.(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 𝑎₀ and 𝑎₁, 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 𝑎_𝑛=𝑎¹(𝛼^𝑛−𝛽^𝑛)−𝑎₀𝛼𝛽𝛼^𝑛−1−𝛽^𝑛−1=𝑎𝛼−𝛽₁(𝛼^𝑛−𝛽^𝑛)+𝑎₀𝑞𝛼^𝑛−1−𝛽^𝑛−1𝛼−𝛽,(2.1) where the last step is due to 𝛼 and 𝛽 being solutions of 𝑡²−𝑝𝑡−𝑞=0. Noting that 𝛼²−𝑝𝛼=𝛼²−(𝛼+𝛽)𝛼=−𝛼𝛽=𝑞 and 𝛼(𝛼−𝑝)=−𝛼𝛽=𝛽(𝛽−𝑝), we may further write the above last expression of 𝑎_𝑛 as 𝑎_𝑛=𝑎¹(𝛼^𝑛−𝛽^𝑛)+𝑎₀𝛼²−𝑝𝛼𝛼^𝑛−1−𝛽^𝑛−1=𝑎𝛼−𝛽₁(𝛼^𝑛−𝛽^𝑛)+𝑎₀𝛼²−𝑝𝛼𝛼^𝑛−1−𝑎₀𝛽²−𝑝𝛽𝛽^𝑛−1=𝑎𝛼−𝛽₀𝛼^𝑛+1−𝛽^𝑛+1+𝑎₁−𝑎₀𝑝(𝛼^𝑛−𝛽^𝑛)𝛼−𝛽.(2.2)

Denote 𝑟(𝑥)=𝑥+√𝑥²−𝐶𝑦 and 𝑠(𝑥)=𝑥−√𝑥²−𝐶𝑦. Comparing expressions (2.2) and (1.6), we have reason to consider the following transform: for a nonzero real or complex number 𝑘, we set 𝛼∶=𝑟(𝑥)𝑘,𝛽∶=𝑠(𝑘𝑥)(2.3) for a certain 𝑥 depending on 𝛼, 𝛽, and 𝑘, which we will find out later. Denote 𝛼+𝛽=𝑝 and 𝛼𝛽=−𝑞; that is, 𝛼 and 𝛽 are roots of 𝑡²−𝑝𝑡−𝑞. By adding the two equations in (2.3) side by side, we obtain 2𝑥=𝑘𝑝. Thus, when 𝑥=𝑘𝑝/2, the equations in (2.2) hold. Meanwhile, by using (𝛼−𝛽)²=(𝛼+𝛽)²−4𝛼𝛽=𝑝²+4𝑞, we have 𝑟(𝑥)−𝑠(𝑥)=2√𝑥²−𝐶𝑦=𝑘(𝛼−𝛽)=𝑘√𝑝²+4𝑞,(2.4) where 𝑥=𝑘𝑝/2. Therefore, we obtain 2𝑘2𝑝²−𝐶𝑦=𝑘√𝑝²+4𝑞,(2.5) which implies 𝑘=±𝐶𝑦−𝑞.(2.6) We first consider the case of 𝑘=√−𝐶𝑦/𝑞.

We now substitute 𝑟(𝑥)=𝑘𝛼, 𝑠(𝑥)=𝑘𝛽, 𝑥=𝑘𝑝/2, and 𝑘=√−𝐶𝑦/𝑞 into (2.2) and simplify as follows:𝑎_𝑛=𝑎⁰(𝑟(𝑥)/𝑘)^𝑛+1−(𝑠(𝑥)/𝑘)^𝑛+1+𝑎₁−𝑎₀𝑝((𝑟(𝑥)/𝑘)^𝑛−(𝑠(𝑥)/𝑘)^𝑛)=𝑎(1/𝑘)(𝑟(𝑥)−𝑠(𝑥))₀𝑟^𝑛+1(𝑥)−𝑠^𝑛+1(𝑥)+𝑘𝑎₁−𝑎₀𝑝(𝑟^𝑛(𝑥)−𝑠^𝑛(𝑥))=𝑎₀𝑘^𝑛(𝑟(𝑥)−𝑠(𝑥))𝐶^𝑛+2−𝑞𝐶𝑦^𝑛𝑃^{𝑛1,𝑦,𝐶}𝑘2𝑝+𝑎¹−𝑎₀𝑝𝐶^𝑛+1−𝑞𝐶𝑦^𝑛−1𝑃^{1,𝑦,𝐶𝑛−1}=𝑎₀𝑘2𝑝𝐶^𝑛+2−𝑞𝐶𝑦^𝑛𝑃^{𝑛1,𝑦,𝐶}2𝑝𝐶𝑦−𝑞+𝑎¹−𝑎₀𝑝𝐶^𝑛+1−𝑞𝐶𝑦^𝑛−1𝑃^{1,𝑦,𝐶𝑛−1}2𝑝𝐶𝑦−𝑞.(2.7) Similarly, for 𝑘=−√−𝐶𝑦/𝑞, we have 𝑎_𝑛=𝑎₀𝐶^𝑛+2−−𝑞𝐶𝑦^𝑛𝑃^{𝑛1,𝑦,𝐶}−2𝑝𝐶𝑦+𝑎−𝑞₁−𝑎₀𝑝𝐶^𝑛+1−−𝑞𝐶𝑦^𝑛−1𝑃^{1,𝑦,𝐶𝑛−1}−2𝑝𝐶𝑦−𝑞.(2.8)

Therefore, we obtain our main result.

Theorem 2.1. Let sequence {𝑎_𝑛} be defined by 𝑎_𝑛=𝑝𝑎_𝑛−1+𝑞𝑎_𝑛−2 (𝑛≥2) with initial conditions 𝑎₀ and 𝑎₁. Then, 𝑎_𝑛 can be presented as (2.7) and (2.8). In particular, for (𝑦,𝐶)=(1,1),(−1,1),(2,1), and (2𝑎,2)(𝑎≠0), respectively, one has 𝑎_𝑛=𝑎₀√−𝑞^𝑛𝑈_𝑛2√𝑝−𝑞+𝑎¹−𝑎₀𝑝√−𝑞^𝑛−1𝑈_𝑛−1𝑎2√𝑝−𝑞_𝑛,=𝑎₀√𝑞^𝑛𝑃_𝑛+12√𝑝𝑞+𝑎¹−𝑎₀𝑝√𝑞^𝑛−1𝑃_𝑛𝑎_𝑛2√𝑝𝑞,=𝑎₀√𝑞^𝑛𝐹_𝑛+1√𝑝𝑞+𝑎¹−𝑎₀𝑝√𝑞^𝑛−1𝐹_𝑛𝑎_𝑛√𝑝𝑞,=𝑎₀−𝑞2^𝑛Φ_𝑛+1𝑝2−𝑞+𝑎¹−𝑎₀𝑝−𝑞2^𝑛−1Φ_𝑛𝑝2𝑎−𝑞_𝑛,=𝑎₀2^𝑛+2−𝑞4𝑎^𝑛𝐷_𝑛𝑝𝑎+𝑎−𝑞,𝑎₁−𝑎₀𝑝2^𝑛+1−𝑞4𝑎^𝑛−1𝐷_𝑛−1𝑝𝑎𝑎−𝑞,𝑎_𝑛,=𝑎₀−√−𝑞^𝑛𝑈_𝑛2√−𝑝−𝑞+𝑎¹−a₀𝑝−√−𝑞^𝑛−1𝑈_𝑛−1𝑎2√−𝑝−𝑞_𝑛,=𝑎₀−√𝑞^𝑛𝑃_𝑛+12−√𝑝𝑞+𝑎¹−𝑎₀𝑝−√𝑞^𝑛−1𝑃_𝑛𝑎_𝑛2−√𝑝𝑞,=𝑎₀−√𝑞^𝑛𝐹_𝑛+1√−𝑝𝑞+𝑎¹−𝑎₀𝑝−√𝑞^𝑛−1𝐹_𝑛𝑎_𝑛√−𝑝𝑞,=𝑎₀−−𝑞2^𝑛Φ_𝑛+1−𝑝2−𝑞+𝑎¹−𝑎₀𝑝−−𝑞2^𝑛−1Φ_𝑛−𝑝2𝑎−𝑞_𝑛,=𝑎₀2^𝑛+2−−𝑞4𝑎^𝑛𝐷_𝑛−𝑝𝑎+𝑎−𝑞,𝑎₁−𝑎₀𝑝2^𝑛+1−−𝑞4𝑎^𝑛−1𝐷_𝑛−1−𝑝𝑎−𝑞,𝑎,(2.9) where 𝑈_𝑛(𝑥), 𝑃_𝑛(𝑥), 𝐹_𝑛(𝑥), Φ_𝑛(𝑥), and 𝐷_𝑛(𝑥,𝑎) 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 𝑎₀ and 𝑎₁, we have the following corollaries.

Corollary 2.2. Let sequence {𝑎_𝑛} be defined by 𝑎_𝑛=𝑝𝑎_𝑛−1+𝑞𝑎_𝑛−2 (𝑛≥2) with initial conditions 𝑎₀=0 and 𝑎₁=𝑑. Then 𝑎_𝑛=𝑑√−𝑞^𝑛−1𝑈_𝑛−1𝑎2√𝑝−𝑞_𝑛,=𝑑√𝑞^𝑛−1𝑃_𝑛𝑎_𝑛2√𝑝𝑞,=𝑑√𝑞^𝑛−1𝐹_𝑛𝑎_𝑛√𝑝𝑞,=𝑑−𝑞2^𝑛−1Φ_𝑛𝑝2𝑎−𝑞_𝑛,=𝑑2^𝑛+1−𝑞4𝑎^𝑛−1𝐷_𝑛−1𝑝𝑎𝑎−𝑞,𝑎_𝑛,=𝑑−√−𝑞^𝑛−1𝑈_𝑛−1𝑎2√−𝑝−𝑞_𝑛,=𝑑−√𝑞^𝑛−1𝑃_𝑛𝑎_𝑛2−√𝑝𝑞,=𝑑−√𝑞^𝑛−1𝐹_𝑛𝑎_𝑛√−𝑝𝑞,=𝑑−−𝑞2^𝑛−1Φ_𝑛−𝑝2𝑎−𝑞_𝑛,=𝑑2^𝑛+1−−𝑞4𝑎^𝑛−1𝐷_𝑛−1−𝑝𝑎−𝑞,𝑎.(2.10)

Corollary 2.3. Let sequence {𝑎_𝑛} be defined by 𝑎_𝑛=𝑝𝑎_𝑛−1+𝑞𝑎_𝑛−2 (𝑛≥2) with initial conditions 𝑎₀=𝑐 and 𝑎₁=𝑝𝑐. Then 𝑎_𝑛=𝑐√−𝑞^𝑛𝑈_𝑛𝑎2√𝑝−𝑞_𝑛,=𝑐√𝑞^𝑛𝑃_𝑛+1𝑎_𝑛2√𝑝𝑞,=𝑐√𝑞^𝑛𝐹_𝑛+1𝑎_𝑛√𝑝𝑞,=𝑐−𝑞2^𝑛Φ_𝑛+1𝑝2𝑎−𝑞_𝑛,=𝑐2^𝑛+2−𝑞4𝑎^𝑛𝐷_𝑛𝑝𝑎𝑎−𝑞,𝑎_𝑛,=𝑐−√−𝑞^𝑛𝑈_𝑛𝑎2√−𝑝−𝑞_𝑛,=𝑐−√𝑞^𝑛𝑃_𝑛+1𝑎_𝑛2−√𝑝𝑞,=𝑐−√𝑞^𝑛𝐹_𝑛+1𝑎_𝑛√−𝑝𝑞,=𝑐−−𝑞2^𝑛Φ_𝑛+1−𝑝2𝑎−𝑞_𝑛,=𝑐2^𝑛+2−−𝑞4𝑎^𝑛𝐷_𝑛−𝑝𝑎−𝑞,𝑎.(2.11)

If 𝑎₁=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=𝑞=1, then 𝑎_𝑛 are the Fibonacci numbers 𝐹_𝑛. Thus, 𝐹_𝑛=(𝑖)^𝑛−1𝑈_𝑛−112𝑖=(𝑖)^𝑛−1𝑈_𝑛−1−𝑖𝐹_𝑛2,=𝑃_𝑛1𝐹_𝑛2,𝐹=𝐹_𝑛(1),_𝑛=𝑖√2^𝑛−1𝐹Φ_𝑛−√2𝑖_𝑛,=2^𝑛+1√𝑖4𝑎^𝑛−1𝐹𝐷_𝑛−1−√𝑎𝑖,𝑎_𝑛,=(−𝑖)^𝑛−1𝑈_𝑛−1𝑖𝐹_𝑛2,=(−1)^𝑛−1𝑃_𝑛−1𝐹_𝑛2,𝐹=(−1)^𝑛−1𝐹_𝑛(−1),_𝑛=−𝑖√2^𝑛−1𝐹Φ_𝑛√2𝑖_𝑛,=2^𝑛+1−√𝑖4𝑎^𝑛−1𝐷_𝑛−1√𝑎𝑖,𝑎,(2.12) where 𝐹_𝑛=(𝑖)^𝑛−1𝑈_𝑛−1(−𝑖/2) was shown in [8] and 𝐹_𝑛=(−𝑖)^𝑛−1𝑈_𝑛−1(𝑖/2) was given by Chen and Louck in [17]. From the above expressions of 𝐹_𝑛, we may obtain many identities. For instance, we have 𝑃_𝑛12=(−1)^𝑛−1𝑃_𝑛−12=𝐹^𝑛(𝑖)_𝑛−1(1)=(−1)^𝑛−1𝐹_𝑛(−1),𝑈_𝑛−1−𝑖2=(−𝑖)^𝑛−1𝑈_𝑛−1𝑖2=√𝑖2^𝑛−1Φ_𝑛−√2𝑖=−𝑖√2^𝑛−1Φ_𝑛√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 𝑎₀=2 and 𝑎₁.

Corollary 2.4. Let sequence {𝑎_𝑛} be defined by 𝑎_𝑛=𝑝𝑎_𝑛−1+𝑞𝑎_𝑛−2 (𝑛≥2) with initial conditions 𝑎₀=2 and 𝑎₁=𝑝. Then 𝑎_𝑛=2√−𝑞^𝑛𝑈_𝑛2√𝑝−𝑞−𝑝√−𝑞^𝑛−1𝑈_𝑛−1𝑎2√𝑝−𝑞_𝑛,=2√𝑞^𝑛𝑃_𝑛+12√𝑝𝑞−𝑝√𝑞^𝑛−1𝑃_𝑛𝑎_𝑛2√𝑝𝑞,=2√𝑞^𝑛𝐹_𝑛+1√𝑝𝑞−𝑝√𝑞^𝑛−1𝐹_𝑛𝑎_𝑛√𝑝𝑞,=2−𝑞2^𝑛Φ_𝑛+1𝑝2−𝑞−𝑝−𝑞2^𝑛−1Φ_𝑛𝑝2𝑎−𝑞_𝑛,=2^𝑛+3−𝑞4𝑎^𝑛𝐷_𝑛𝑝𝑎−𝑞,𝑎−𝑝2^𝑛+1−𝑞4𝑎^𝑛−1𝐷_𝑛−1𝑝𝑎𝑎−𝑞,𝑎_𝑛,=2−√−𝑞^𝑛𝑈_𝑛2√−𝑝−𝑞−𝑝−√−𝑞^𝑛−1𝑈_𝑛−1𝑎2√−𝑝−𝑞_𝑛,=2−√𝑞^𝑛𝑃_𝑛+12−√𝑝𝑞−𝑝−√𝑞^𝑛−1𝑃_𝑛𝑎_𝑛2−√𝑝𝑞,=2−√𝑞^𝑛𝐹_𝑛+1√−𝑝𝑞−𝑝−√𝑞^𝑛−1𝐹_𝑛𝑎_𝑛√−𝑝𝑞,=2−−𝑞2^𝑛Φ_𝑛+1−𝑝2−𝑞−𝑝−−𝑞2^𝑛−1Φ_𝑛−𝑝2𝑎−𝑞_𝑛,=2^𝑛+3−−𝑞4𝑎^𝑛𝐷_𝑛−𝑝𝑎−𝑞,𝑎−𝑝2^𝑛+1−−𝑞4𝑎^𝑛−1𝐷_𝑛−1−𝑝𝑎−𝑞,𝑎.(2.14) In addition, one has 𝑎_𝑛=2√−𝑞^𝑛𝑇_𝑛𝑎2√𝑝−𝑞,(2.15)_𝑛=2−√−𝑞^𝑛𝑇_𝑛−2√𝑝−𝑞,(2.16) where 𝑇_𝑛(𝑥) 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 𝑈_𝑛(𝑥)=2𝑥𝑈_𝑛−1(𝑥)−𝑈_𝑛−2(𝑥), one easily sees 𝑎_𝑛=√−𝑞^𝑛=√2𝑈^𝑛2√𝑝−𝑞−√𝑝−𝑞𝑈^𝑛−12√𝑝−𝑞−𝑞^𝑛=√2𝑈^𝑛2√𝑝−𝑞−𝑈^𝑛2√𝑝−𝑞+𝑈^𝑛−22√𝑝−𝑞−𝑞^𝑛𝑈^𝑛2√𝑝−𝑞−𝑈^𝑛−22√𝑝−𝑞.(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), 𝑈_𝑛(𝑥)−𝑈_𝑛−2(𝑥)=2𝑇_𝑛(𝑥), the last expression of 𝑎_𝑛 implies (2.15). Equation (2.16) can be proved similarly.

As an example, the Lucas number sequence {𝐿_𝑛} defined by (1.1) with 𝑝=𝑞=1 and initial conditions 𝐿₀=2 and 𝐿₁=1 has the explicit formula for its general term: 𝐿_𝑛=2𝑖^𝑛𝑇_𝑛−𝑖2=2(−𝑖)^𝑛𝑇_𝑛𝑖2.(2.18)

3. Examples and Applications

We first give some examples of Corollary 2.2 for sequences {𝑎_𝑛} that are primary solutions of (1.1).

Example 3.1. If 𝑝=2 and 𝑞=1, then 𝑎_𝑛 defined by (1.1) with initial conditions 𝑎₀=0 and 𝑎₁=1 are the Pell numbers 𝑃_𝑛. Thus, from Corollary 2.2, we have 𝑃𝑃_𝑛=(𝑖)^𝑛−1𝑈_𝑛−1(−𝑖)=(−𝑖)^𝑛−1𝑈_𝑛−1(𝑖),_𝑛𝑃=𝑃_𝑛(1)=(−1)^𝑛−1𝑃_𝑛(−1),_𝑛𝑃=𝐹_𝑛(2)=(−1)^𝑛−1𝐹_𝑛(−2),_𝑛=𝑖√2^𝑛−1Φ_𝑛−2√2𝑖=−𝑖√2^𝑛−1𝑃Φ_𝑛2√2𝑖_𝑛,=2^𝑛+1√𝑖4𝑎^𝑛−1=2_𝑛+1𝐷_𝑛−1−2√𝑎𝑖,𝑎−√𝑖4𝑎^𝑛−1𝐷_𝑛−12√𝑎𝑖,𝑎.(3.1)

Example 3.2. If 𝑝=1 and 𝑞=2, then 𝑎_𝑛 defined by (1.1) with initial conditions 𝑎₀=0 and 𝑎₁=1 are the Jacobsthal numbers 𝐽_𝑛 (see Bergum et al. [19]). Thus Corollary 2.2 gives the expressions of 𝐽_𝑛 as follows: 𝐽_𝑛=√2𝑖^𝑛−1𝑈_𝑛−12−√𝑖2=−√2𝑖^𝑛−1𝑈_𝑛−1𝐽_𝑛2√𝑖2,=√2^𝑛−1𝑃_𝑛2√12=−√2^𝑛−1𝑃_𝑛𝐽_𝑛−2√12,=√2^𝑛−1𝐹_𝑛√12=−√2^𝑛−1𝐹_𝑛𝐽_𝑛−√12,𝐽=𝑖^𝑛−1Φ_𝑛(−𝑝i)=(−𝑖)^𝑛−1Φ_𝑛(𝑝𝑖),_𝑛=2^𝑛+1√𝑖2𝑎^𝑛−1𝐷_𝑛−1−𝑝√𝑎𝑖=2_𝑛+1√2,𝑎−√𝑖2𝑎^𝑛−1𝐷_𝑛−1𝑝√𝑎𝑖√2,𝑎.(3.2)

Example 3.3. If 𝑝=3 and 𝑞=−2, then 𝑎_𝑛 defined by (1.1) with initial conditions 𝑎₀=0 and 𝑎₁=1 are the Mersenne numbers 𝑀_𝑛=2^𝑛−1. From Corollary 2.2, we have 𝑀_𝑛=√2^𝑛−1𝑈_𝑛−12√32=−√2^𝑛−1𝑈_𝑛−1𝑀_𝑛2−√32,=√2𝑖^𝑛−1𝑃_𝑛−2√3𝑖2=−√2𝑖^𝑛−1𝑃_𝑛𝑀_𝑛2√3𝑖2,=√2𝑖^𝑛−1𝐹_𝑛−√3𝑖2=−√2𝑖^𝑛−1𝐹_𝑛𝑀_𝑛√3𝑖2,𝑀=Φ_𝑛(3)=(−1)^𝑛−1Φ_𝑛(−3),_𝑛=2^𝑛+1√12𝑎^𝑛−1𝐷_𝑛−13√𝑎=2_𝑛+1√2,𝑎−√12𝑎^𝑛−1𝐷_𝑛−1−3√𝑎√2,𝑎.(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 𝑞=1, then 𝑎_𝑛 defined by (1.1) with initial conditions 𝑎₀=2 and 𝑎₁=1 are the Lucas numbers 𝐿_𝑛. Thus, besides (2.18), we have 𝐿_𝑛=2𝑖^𝑛𝑈_𝑛−𝑖2−𝑖^𝑛−1𝑈_𝑛−1−𝑖=2(−𝑖)_𝑛2𝑈_𝑛𝑖2−(−𝑖)^𝑛−1𝑈_𝑛−1𝑖𝐿_𝑛2,=2𝑃_𝑛+11=2(−1)_𝑛2−𝑃^𝑛21𝑃_𝑛+1−12−(−1)^𝑛−1𝑃_𝑛−1𝐿_𝑛2,𝐿=2𝐹_𝑛+1(1)−𝐹_𝑛(1)=2(−1)^𝑛𝐹_𝑛+1(−1)−(−1)^𝑛−1𝐹_𝑛(−1),_𝑛=2𝑖√2^𝑛Φ_𝑛+1−√2𝑖−𝑖√2^𝑛−1=2−𝑖Φ_𝑛−√2𝑖√2^𝑛Φ_𝑛+1√2𝑖−−𝑖√2^𝑛−1𝐿Φ_𝑛√2𝑖_𝑛,=2^𝑛+3√𝑖4𝑎^𝑛𝐷_𝑛−√𝑎𝑖,𝑎−2^𝑛+1√𝑖4𝑎^𝑛−1=2_𝑛+3𝐷_𝑛−1−√𝑎𝑖,𝑎−√𝑖4𝑎^𝑛𝐷_𝑛√𝑎𝑖,𝑎−2^𝑛+1−√𝑖4𝑎^𝑛−1𝐷_𝑛−1√𝑎𝑖,𝑎.(3.4)

Example 3.5. If 𝑝=2 and 𝑞=1, then 𝑎_𝑛 defined by (1.1) with initial conditions 𝑎₀=2 and 𝑎₁=2 are the Pell-Lucas numbers 𝐴_𝑛 (see Example 2 in [11]). Thus, from Corollary 2.4, we obtain 𝐴𝐴_𝑛=2𝑖^𝑛𝑇_𝑛(−𝑖)=2(−𝑖)^𝑛𝑇_𝑛(𝑖),_𝑛𝐴=2𝑖^𝑛𝑈_𝑛(−𝑖)−2𝑖^𝑛−1𝑈_𝑛−1(−𝑖)=2𝑖^𝑛𝑈_𝑛(−𝑖)−2𝑖^𝑛−1𝑈_𝑛−1(−𝑖),_𝑛𝐴=2𝑃_𝑛+1(1)−2𝑃_𝑛(1)=2(−1)^𝑛𝑃_𝑛+1(−1)−𝑝(−1)^𝑛−1𝑃_𝑛(−1),_𝑛𝐴=2𝐹_𝑛+1(2)−2𝐹_𝑛(2)=2(−1)^𝑛𝐹_𝑛+1(−2)−𝑝(−1)^𝑛−1𝐹_𝑛(−2),_𝑛=2−𝑖√2^𝑛Φ_𝑛+12√2𝑖−2−𝑖√2^𝑛−1=2𝑖Φ_𝑛2√2𝑖√2^𝑛Φ_𝑛+1−2√2𝑖−2𝑖√2^𝑛−1𝐴Φ_𝑛−2√2𝑖_𝑛,=2^𝑛+3√𝑖4𝑎^𝑛𝐷_𝑛−2√𝑎𝑖,𝑎−2^𝑛+2√𝑖4𝑎^𝑛−1=2_𝑛+3𝐷_𝑛−1−2√𝑎𝑖,𝑎−√𝑖4𝑎^𝑛𝐷_𝑛2√𝑎𝑖,𝑎−2^𝑛+2−√𝑖4𝑎^𝑛−1𝐷_𝑛−12√𝑎𝑖,𝑎.(3.5)

Example 3.6. If 𝑝=1 and 𝑞=2, then 𝑎_𝑛 defined by (1.1) with initial conditions 𝑎₀=2 and 𝑎₁=1 are the Jacobsthal-Lucas numbers 𝐵_𝑛 (see Example 2 in [11]). Thus, 𝐵_𝑛=2√2𝑖^𝑛𝑇_𝑛−2√𝑖2=2−√2𝑖^𝑛𝑇_𝑛𝐵_𝑛2√𝑖2,=2√2𝑖^𝑛𝑈_𝑛−2√𝑖2−√2𝑖^𝑛−1𝑈_𝑛−1=2−√−2√𝑖22𝑖^𝑛𝑈_𝑛2√𝑖2−−√2𝑖^𝑛−1𝑈_𝑛−1𝐵_𝑛2√𝑖2,=2√2^𝑛𝑃_𝑛+12√12−√2^𝑛−1𝑃_𝑛=2−√2√122^𝑛𝑃_𝑛+1−2√12−−√2^𝑛−1𝑃_𝑛𝐵_𝑛−2√12,=2√2^𝑛𝐹_𝑛+1√12−√2^𝑛−1𝐹_𝑛=2−√√122^𝑛𝐹_𝑛+1−√12−−√2^𝑛−1𝐹_𝑛𝐵_𝑛−√12,𝐵=2𝑖^𝑛Φ_𝑛+1(−𝑖)−𝑖^𝑛−1Φ_𝑛(−𝑖)=2(−𝑖)^𝑛Φ_𝑛+1(𝑖)−(−𝑖)^𝑛−1Φ_𝑛(𝑖),_𝑛=2^𝑛+3√𝑖2𝑎^𝑛𝐷_𝑛−√𝑎𝑖√2,𝑎−2^𝑛+1√𝑖2𝑎^𝑛−1𝐷_𝑛−1−√𝑎𝑖=2_𝑛+3√2,𝑎−√𝑖2𝑎^𝑛𝐷_𝑛√𝑎𝑖√2,𝑎−2^𝑛+1−√𝑖2𝑎^𝑛−1𝐷_𝑛−1√𝑎𝑖√2,𝑎.(3.6)

Example 3.7. If 𝑝=3 and 𝑞=−2, then 𝑎_𝑛 defined by (1.1) with initial conditions 𝑎₀=2 and 𝑎₁=3 are the Fermat numbers 𝑓_𝑛 (see [20]). Thus, from Corollary 2.4, we obtain 𝑓_𝑛=2√2^𝑛𝑇_𝑛2√32=2−√2^𝑛𝑇_𝑛𝑓_𝑛−2√32,=2√2^𝑛𝑈_𝑛2√32−3√2^𝑛−1𝑈_𝑛−1=2−√2√322^𝑛𝑈_𝑛−2√32−3−√2^𝑛−1𝑈_𝑛−1𝑓_𝑛−2√32,=2√2𝑖^𝑛𝑃_𝑛+1−2√3𝑖2−3√2𝑖^𝑛−1𝑃_𝑛=2−√−2√3𝑖22𝑖^𝑛𝑃_𝑛+12√3𝑖2−3−√2𝑖^𝑛−1𝑃_𝑛𝑓_𝑛2√3𝑖2,=2√2𝑖^𝑛𝐹_𝑛+1−√3𝑖2−3√2𝑖^𝑛−1𝐹_𝑛=2−√−√3𝑖22𝑖^𝑛𝐹_𝑛+1√3𝑖2−3−√2𝑖^𝑛−1𝐹_𝑛𝑓_𝑛√3𝑖2,𝑓=2Φ_𝑛+1(3)−3Φ_𝑛(3)=2(−1)^𝑛Φ_𝑛+1(−3)−3(−1)^𝑛−1Φ_𝑛(−3),_𝑛=2^𝑛+3√12𝑎^𝑛𝐷_𝑛3√𝑎√2,𝑎−(3)2^𝑛+1√12𝑎^𝑛−1𝐷_𝑛−13√𝑎=2_𝑛+3√2,𝑎−√12𝑎^𝑛𝐷_𝑛−3√𝑎√2,𝑎−(3)2^𝑛+1−√12𝑎^𝑛−1𝐷_𝑛−1−3√𝑎√2,𝑎.(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: 𝑃^{𝑛1,𝑦,𝐶}(𝑥)=𝛼(𝑥)𝑃^{1,𝑦,𝐶𝑛−1}(𝑥)+𝐶⁻²(2𝑥−𝛼(𝑥)𝐶)(𝛽(𝑥))^𝑛−1,(3.8) where 𝑃^{𝑛1,𝑦,𝐶}(𝑥) satisfies the recurrence relation of order 2, 𝑃^{𝑛1,𝑦,𝐶}=𝑝𝑃^{1,𝑦,𝐶𝑛−1}+𝑞𝑃^{1,𝑦,𝐶𝑛−2} with coefficients 𝑝(𝑥) and 𝑞(𝑥), and 𝛼(𝑥)+𝛽(𝑥)=𝑝(𝑥) and 𝛼(𝑥)𝛽(𝑥)=−𝑞(𝑥). Clearly (see (19) and (20) in [11]), 𝛼=1𝐶𝑥+√𝛽=1𝑥²−𝐶𝑦,𝐶𝑥−√𝑥²−𝐶𝑦.(3.9) For 𝑦=−1 and 𝐶=1, we have 𝑃^𝑛1,−1,1(𝑥)=𝐹_𝑛+1(2𝑥), where 𝐹_𝑛(𝑥) are the Fibonacci polynomials, and we can write (3.8) as 𝐹_𝑛+1(2𝑥)=𝛼(𝑥)𝐹_𝑛(2𝑥)+(2𝑥−𝛼(𝑥))(𝛽(𝑥))^𝑛−1=𝛼(𝑥)𝐹_𝑛(2𝑥)+(𝛽(𝑥))^𝑛,(3.10) where 𝛼(𝑥)=𝑥+√𝑥²+1 and 𝛽(𝑥)=𝑥−√𝑥²+1. If 𝑥=1/2, then 𝐹_𝑛(1)=𝐹_𝑛, the Fibonacci numbers, and 𝛼12=1+√52,𝛽21=1−√52.(3.11)

Thus (3.10) yields the identity 𝐹_𝑛+1=1+√52𝐹^𝑛+1−√52^𝑛,(3.12) or equivalently, 1−√52𝐹^𝑛+1+𝐹_𝑛=1−√52^𝑛+1.(3.13)

Similarly, if 𝑥=1, then 𝐹_𝑛(2)=𝑃_𝑛, the Pell numbers, and 𝛼(1)=1+√2,𝛽(1)=1−√2.(3.14) Thus (3.10) yields the identity𝑃_𝑛+1=1+√2𝑃_𝑛+1−√2^𝑛,(3.15) or equivalently, 1−√2𝑃_𝑛+1+𝑃_𝑛=1−√2^𝑛+1.(3.16)

Substituting 𝑥=1/(2√2) into (3.10) and noting 𝐹_𝑛(1/√2)=𝐽_𝑛/(√2)^𝑛, where 𝐽_𝑛 are the Jacobsthal numbers, we obtain the identity 𝐽_𝑛+1−2𝐽_𝑛=(−1)^𝑛.(3.17) When 𝑥=−3𝑖/(2√2), 𝐹_𝑛(−3𝑖/(2√2))=𝑀_𝑛/(√2𝑖)^𝑛−1, the Mersenne numbers. Hence (3.10) gives 𝑀_𝑛+1−𝑀_𝑛=2^𝑛.

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.