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ISRN Discrete Mathematics
Volume 2011 (2011), Article ID 674167, 16 pages
http://dx.doi.org/10.5402/2011/674167
Research Article

Sequences of Numbers Meet the Generalized Gegenbauer-Humbert Polynomials

1Department of Mathematics and Computer Science, Illinois Wesleyan University, Bloomington, IL 61702, USA
2Department of Mathematical Sciences, University of Nevada Las Vegas, Las Vegas, NV 89154-4020, USA
3Department of Electrical Engineering, National Taiwan University, Taipei 106, Taiwan

Received 6 July 2011; Accepted 25 August 2011

Academic Editor: W. Liu

Copyright © 2011 Tian-Xiao He et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 {𝑎𝑛} is called sequence of order 2 if it satisfies the linear recurrence relation of order 2: 𝑎𝑛=𝑝𝑎𝑛1+𝑞𝑎𝑛2,𝑛2,(1.1) for some nonzero constants 𝑝 and 𝑞 and initial conditions 𝑎0 and 𝑎1. In Mansour [1], the sequence {𝑎𝑛}𝑛0 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, 𝑎𝑛 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 𝐹𝑛=𝑖𝑛𝑈𝑛(𝑖/2) and 𝐿𝑛=2𝑖𝑛𝑇𝑛(𝑖/2), where 𝐹𝑛 and 𝐿𝑛, respectively, are the Fibonacci numbers and Lucas numbers, and 𝑇𝑛 and 𝑈𝑛 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 {𝑎𝑛} generated by (1.1). For the sake of the reader's convenience, we cite this result as follows.

Proposition 1.1 (see [11]). Let {𝑎𝑛} be a sequence of order 2 satisfying linear recurrence relation (1.1), and let 𝛼 and 𝛽 be two roots of of quadratic equation 𝑥2𝑝𝑥𝑞=0. Then 𝑎𝑛=𝑎1𝛽𝑎0𝛼𝛼𝛽𝑛𝑎1𝛼𝑎0𝛽𝛼𝛽𝑛,if𝛼𝛽,𝑛𝑎1𝛼𝑛1(𝑛1)𝑎0𝛼𝑛,if𝛼=𝛽.(1.2)

A sequence of the generalized Gegenbauer-Humbert polynomials {𝑃𝑛𝜆,𝑦,𝐶(𝑥)}𝑛0 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)𝐶2𝑥𝑡+𝑦𝑡2𝜆=𝑛0𝑃𝑛𝜆,𝑦,𝐶(𝑥)𝑡𝑛,(1.3) where 𝜆>0, 𝑦 and 𝐶0 are real numbers. As special cases of (1.3), we consider 𝑃𝑛𝜆,𝑦,𝐶(𝑥) as follows (see [14]): 𝑃𝑛1,1,1(𝑥)=𝑈𝑛(𝑥), the Chebyshev polynomial of the second kind,𝑃𝑛1/2,1,1(𝑥)=𝜓𝑛(𝑥), the Legendre polynomial,𝑃𝑛1,1,1(𝑥)=𝑃𝑛+1(𝑥), the Pell polynomial,𝑃𝑛1,1,1(𝑥/2)=𝐹𝑛+1(𝑥), the Fibonacci polynomial,𝑃𝑛1,2,1(𝑥/2)=Φ𝑛+1(𝑥), the Fermat polynomial of the first kind,𝑃𝑛1,2𝑎,2(𝑥)=𝐷𝑛(𝑥,𝑎), the Dickson polynomial of the second kind, 𝑎0 (see, e.g., [13]),

where 𝑎 is a real parameter, and 𝐹𝑛=𝐹𝑛(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 𝑃𝑛𝜆,𝑦,𝐶(𝑥)=2𝑥𝜆+𝑛1𝑃𝐶𝑛𝜆,𝑦,𝐶𝑛1(𝑥)𝑦2𝜆+𝑛2𝑃𝐶𝑛𝜆,𝑦,𝐶𝑛2(𝑥),(1.4) for all 𝑛2 with initial conditions𝑃0𝜆,𝑦,𝐶(𝑥)=Φ(0)=𝐶𝜆,𝑃1𝜆,𝑦,𝐶(𝑥)=Φ(0)=2𝜆𝑥𝐶𝜆1,(1.5) the following theorem has been obtained in [11].

Theorem 1.2 (see [11]). Let 𝑥±𝐶𝑦. The generalized Gegenbauer-Humbert polynomials {𝑃𝑛1,𝑦,𝐶(𝑥)}𝑛0 defined by expansion (1.3) can be expressed as 𝑃𝑛1,𝑦,𝐶(𝑥)=𝐶𝑛2𝑥+𝑥2𝐶𝑦𝑛+1𝑥𝑥2𝐶𝑦𝑛+12𝑥2𝐶𝑦.(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 𝛼𝛽, the first formula in (1.2) can be written as 𝑎𝑛=𝑎1(𝛼𝑛𝛽𝑛)𝑎0𝛼𝛼𝛽𝑛1𝛽𝑛1=𝑎𝛼𝛽1(𝛼𝑛𝛽𝑛)+𝑎0𝑞𝛼𝑛1𝛽𝑛1,𝛼𝛽(2.1) where the last step is due to 𝛼 and 𝛽 being solutions of 𝑡2𝑝𝑡𝑞=0. Noting that 𝛼2𝑝𝛼=𝛼2(𝛼+𝛽)𝛼=𝛼𝛽=𝑞 and 𝛼(𝛼𝑝)=𝛼𝛽=𝛽(𝛽𝑝), we may further write the above last expression of 𝑎𝑛 as 𝑎𝑛=𝑎1(𝛼𝑛𝛽𝑛)+𝑎0𝛼2𝛼𝑝𝛼𝑛1𝛽𝑛1=𝑎𝛼𝛽1(𝛼𝑛𝛽𝑛)+𝑎0𝛼2𝛼𝑝𝛼𝑛1𝑎0𝛽2𝛽𝑝𝛽𝑛1=𝑎𝛼𝛽0𝛼𝑛+1𝛽𝑛+1+𝑎1𝑎0𝑝(𝛼𝑛𝛽𝑛).𝛼𝛽(2.2)

Denote 𝑟(𝑥)=𝑥+𝑥2𝐶𝑦 and 𝑠(𝑥)=𝑥𝑥2𝐶𝑦. 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(𝑥)𝑘(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 𝑡2𝑝𝑡𝑞. 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 (𝛼𝛽)2=(𝛼+𝛽)24𝛼𝛽=𝑝2+4𝑞, we have 𝑟(𝑥)𝑠(𝑥)=2𝑥2𝐶𝑦=𝑘(𝛼𝛽)=𝑘𝑝2+4𝑞,(2.4) where 𝑥=𝑘𝑝/2. Therefore, we obtain 2𝑘𝑝22𝐶𝑦=𝑘𝑝2+4𝑞,(2.5) which implies 𝑘=±𝐶𝑦𝑞.(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:𝑎𝑛=𝑎0(𝑟(𝑥)/𝑘)𝑛+1(𝑠(𝑥)/𝑘)𝑛+1+𝑎1𝑎0𝑝((𝑟(𝑥)/𝑘)𝑛(𝑠(𝑥)/𝑘)𝑛)=𝑎(1/𝑘)(𝑟(𝑥)𝑠(𝑥))0𝑟𝑛+1(𝑥)𝑠𝑛+1𝑎(𝑥)+𝑘1𝑎0𝑝(𝑟𝑛(𝑥)𝑠𝑛(𝑥))𝑘𝑛(𝑟(𝑥)𝑠(𝑥))=𝑎0𝐶𝑛+2𝑞𝐶𝑦𝑛𝑃𝑛1,𝑦,𝐶𝑘𝑝2+𝑎1𝑎0𝑝𝐶𝑛+1𝑞𝐶𝑦𝑛1𝑃1,𝑦,𝐶𝑛1𝑘𝑝2=𝑎0𝐶𝑛+2𝑞𝐶𝑦𝑛𝑃𝑛1,𝑦,𝐶𝑝2𝐶𝑦+𝑎𝑞1𝑎0𝑝𝐶𝑛+1𝑞𝐶𝑦𝑛1𝑃1,𝑦,𝐶𝑛1𝑝2𝐶𝑦.𝑞(2.7) Similarly, for 𝑘=𝐶𝑦/𝑞, we have 𝑎𝑛=𝑎0𝐶𝑛+2𝑞𝐶𝑦𝑛𝑃𝑛1,𝑦,𝐶𝑝2𝐶𝑦+𝑎𝑞1𝑎0𝑝𝐶𝑛+1𝑞𝐶𝑦𝑛1𝑃1,𝑦,𝐶𝑛1𝑝2𝐶𝑦.𝑞(2.8)

Therefore, we obtain our main result.

Theorem 2.1. Let sequence {𝑎𝑛} be defined by 𝑎𝑛=𝑝𝑎𝑛1+𝑞𝑎𝑛2 (𝑛2) with initial conditions 𝑎0 and 𝑎1. Then, 𝑎𝑛 can be presented as (2.7) and (2.8). In particular, for (𝑦,𝐶)=(1,1),(1,1),(2,1), and (2𝑎,2)(𝑎0), respectively, one has 𝑎𝑛=𝑎0𝑞𝑛𝑈𝑛𝑝2+𝑎𝑞1𝑎0𝑝𝑞𝑛1𝑈𝑛1𝑝2,𝑎𝑞𝑛=𝑎0𝑞𝑛𝑃𝑛+1𝑝2𝑞+𝑎1𝑎0𝑝𝑞𝑛1𝑃𝑛𝑝2𝑞,𝑎𝑛=𝑎0𝑞𝑛𝐹𝑛+1𝑝𝑞+𝑎1𝑎0𝑝𝑞𝑛1𝐹𝑛𝑝𝑞,𝑎𝑛=𝑎0𝑞2𝑛Φ𝑛+1𝑝2+𝑎𝑞1𝑎0𝑝𝑞2𝑛1Φ𝑛𝑝2,𝑎𝑞𝑛=𝑎02𝑛+2𝑞4𝑎𝑛𝐷𝑛𝑝𝑎+𝑎𝑞,𝑎1𝑎0𝑝2𝑛+1𝑞4𝑎𝑛1𝐷𝑛1𝑝𝑎,𝑎𝑞,𝑎𝑛=𝑎0𝑞𝑛𝑈𝑛𝑝2+𝑎𝑞1a0𝑝𝑞𝑛1𝑈𝑛1𝑝2,𝑎𝑞𝑛=𝑎0𝑞𝑛𝑃𝑛+1𝑝2𝑞+𝑎1𝑎0𝑝𝑞𝑛1𝑃𝑛𝑝2𝑞,𝑎𝑛=𝑎0𝑞𝑛𝐹𝑛+1𝑝𝑞+𝑎1𝑎0𝑝𝑞𝑛1𝐹𝑛𝑝𝑞,𝑎𝑛=𝑎0𝑞2𝑛Φ𝑛+1𝑝2+𝑎𝑞1𝑎0𝑝𝑞2𝑛1Φ𝑛𝑝2,𝑎𝑞𝑛=𝑎02𝑛+2𝑞4𝑎𝑛𝐷𝑛𝑝𝑎+𝑎𝑞,𝑎1𝑎0𝑝2𝑛+1𝑞4𝑎𝑛1𝐷𝑛1𝑝𝑎,𝑞,𝑎(2.9) where 𝑈𝑛(𝑥), 𝑃𝑛(𝑥), 𝐹𝑛(𝑥), Φ𝑛(𝑥), and 𝐷𝑛(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 {𝑎𝑛} be defined by 𝑎𝑛=𝑝𝑎𝑛1+𝑞𝑎𝑛2 (𝑛2) with initial conditions 𝑎0=0 and 𝑎1=𝑑. Then 𝑎𝑛=𝑑𝑞𝑛1𝑈𝑛1𝑝2,𝑎𝑞𝑛=𝑑𝑞𝑛1𝑃𝑛𝑝2𝑞,𝑎𝑛=𝑑𝑞𝑛1𝐹𝑛𝑝𝑞,𝑎𝑛=𝑑𝑞2𝑛1Φ𝑛𝑝2,𝑎𝑞𝑛=𝑑2𝑛+1𝑞4𝑎𝑛1𝐷𝑛1𝑝𝑎,𝑎𝑞,𝑎𝑛=𝑑𝑞𝑛1𝑈𝑛1𝑝2,𝑎𝑞𝑛=𝑑𝑞𝑛1𝑃𝑛𝑝2𝑞,𝑎𝑛=𝑑𝑞𝑛1𝐹𝑛𝑝𝑞,𝑎𝑛=𝑑𝑞2𝑛1Φ𝑛𝑝2,𝑎𝑞𝑛=𝑑2𝑛+1𝑞4𝑎𝑛1𝐷𝑛1𝑝𝑎.𝑞,𝑎(2.10)

Corollary 2.3. Let sequence {𝑎𝑛} be defined by 𝑎𝑛=𝑝𝑎𝑛1+𝑞𝑎𝑛2 (𝑛2) with initial conditions 𝑎0=𝑐 and 𝑎1=𝑝𝑐. Then 𝑎𝑛=𝑐𝑞𝑛𝑈𝑛𝑝2,𝑎𝑞𝑛=𝑐𝑞𝑛𝑃𝑛+1𝑝2𝑞,𝑎𝑛=𝑐𝑞𝑛𝐹𝑛+1𝑝𝑞,𝑎𝑛=𝑐𝑞2𝑛Φ𝑛+1𝑝2,𝑎𝑞𝑛=𝑐2𝑛+2𝑞4𝑎𝑛𝐷𝑛𝑝𝑎,𝑎𝑞,𝑎𝑛=𝑐𝑞𝑛𝑈𝑛𝑝2,𝑎𝑞𝑛=𝑐𝑞𝑛𝑃𝑛+1𝑝2𝑞,𝑎𝑛=𝑐𝑞𝑛𝐹𝑛+1𝑝𝑞,𝑎𝑛=𝑐𝑞2𝑛Φ𝑛+1𝑝2,𝑎𝑞𝑛=𝑐2𝑛+2𝑞4𝑎𝑛𝐷𝑛𝑝𝑎.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 𝑎𝑛 are the Fibonacci numbers 𝐹𝑛. Thus, 𝐹𝑛=(𝑖)𝑛1𝑈𝑛112𝑖=(𝑖)𝑛1𝑈𝑛1𝑖2,𝐹𝑛=𝑃𝑛12,𝐹𝑛=𝐹𝑛𝐹(1),𝑛=𝑖2𝑛1Φ𝑛,𝐹2𝑖𝑛=2𝑛+1𝑖4𝑎𝑛1𝐷𝑛1,𝐹𝑎𝑖,𝑎𝑛=(𝑖)𝑛1𝑈𝑛1𝑖2,𝐹𝑛=(1)𝑛1𝑃𝑛12,𝐹𝑛=(1)𝑛1𝐹𝑛𝐹(1),𝑛=𝑖2𝑛1Φ𝑛,𝐹2𝑖𝑛=2𝑛+1𝑖4𝑎𝑛1𝐷𝑛1,𝑎𝑖,𝑎(2.12) where 𝐹𝑛=(𝑖)𝑛1𝑈𝑛1(𝑖/2) was shown in [8] and 𝐹𝑛=(𝑖)𝑛1𝑈𝑛1(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 𝑃𝑛12=(1)𝑛1𝑃𝑛12=𝐹𝑛(1)=(1)𝑛1𝐹𝑛(1),(𝑖)𝑛1𝑈𝑛1𝑖2=(𝑖)𝑛1𝑈𝑛1𝑖2=𝑖2𝑛1Φ𝑛=𝑖2𝑖2𝑛1Φ𝑛,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 {𝑎𝑛} be defined by 𝑎𝑛=𝑝𝑎𝑛1+𝑞𝑎𝑛2 (𝑛2) with initial conditions 𝑎0=2 and 𝑎1=𝑝. Then 𝑎𝑛=2𝑞𝑛𝑈𝑛𝑝2𝑞𝑝𝑞𝑛1𝑈𝑛1𝑝2,𝑎𝑞𝑛=2𝑞𝑛𝑃𝑛+1𝑝2𝑞𝑝𝑞𝑛1𝑃𝑛𝑝2𝑞,𝑎𝑛=2𝑞𝑛𝐹𝑛+1𝑝𝑞𝑝𝑞𝑛1𝐹𝑛𝑝𝑞,𝑎𝑛=2𝑞2𝑛Φ𝑛+1𝑝2𝑞𝑝𝑞2𝑛1Φ𝑛𝑝2,𝑎𝑞𝑛=2𝑛+3𝑞4𝑎𝑛𝐷𝑛𝑝𝑎𝑞,𝑎𝑝2𝑛+1𝑞4𝑎𝑛1𝐷𝑛1𝑝𝑎,𝑎𝑞,𝑎𝑛=2𝑞𝑛𝑈𝑛𝑝2𝑞𝑝𝑞𝑛1𝑈𝑛1𝑝2,𝑎𝑞𝑛=2𝑞𝑛𝑃𝑛+1𝑝2𝑞𝑝𝑞𝑛1𝑃𝑛𝑝2𝑞,𝑎𝑛=2𝑞𝑛𝐹𝑛+1𝑝𝑞𝑝𝑞𝑛1𝐹𝑛𝑝𝑞,𝑎𝑛=2𝑞2𝑛Φ𝑛+1𝑝2𝑞𝑝𝑞2𝑛1Φ𝑛𝑝2,𝑎𝑞𝑛=2𝑛+3𝑞4𝑎𝑛𝐷𝑛𝑝𝑎𝑞,𝑎𝑝2𝑛+1𝑞4𝑎𝑛1𝐷𝑛1𝑝𝑎.𝑞,𝑎(2.14) In addition, one has 𝑎𝑛=2𝑞𝑛𝑇𝑛𝑝2𝑎𝑞,(2.15)𝑛=2𝑞𝑛𝑇𝑛𝑝2𝑞,(2.16) where 𝑇𝑛(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 𝑈𝑛(𝑥)=2𝑥𝑈𝑛1(𝑥)𝑈𝑛2(𝑥), one easily sees 𝑎𝑛=𝑞𝑛2𝑈𝑛𝑝2𝑝𝑞𝑈𝑞𝑛1𝑝2=𝑞𝑞𝑛2𝑈𝑛𝑝2𝑈𝑞𝑛𝑝2𝑞+𝑈𝑛2𝑝2=𝑞𝑞𝑛𝑈𝑛𝑝2𝑞𝑈𝑛2𝑝2.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), 𝑈𝑛(𝑥)𝑈𝑛2(𝑥)=2𝑇𝑛(𝑥), the last expression of 𝑎𝑛 implies (2.15). Equation (2.16) can be proved similarly.

As an example, the Lucas number sequence {𝐿𝑛} defined by (1.1) with 𝑝=𝑞=1 and initial conditions 𝐿0=2 and 𝐿1=1 has the explicit formula for its general term: 𝐿𝑛=2𝑖𝑛𝑇𝑛𝑖2=2(𝑖)𝑛𝑇𝑛𝑖2.(2.18)

3. Examples and Applications

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

Example 3.1. If 𝑝=2 and 𝑞=1, then 𝑎𝑛 defined by (1.1) with initial conditions 𝑎0=0 and 𝑎1=1 are the Pell numbers 𝑃𝑛. Thus, from Corollary 2.2, we have 𝑃𝑛=(𝑖)𝑛1𝑈𝑛1(𝑖)=(𝑖)𝑛1𝑈𝑛1𝑃(𝑖),𝑛=𝑃𝑛(1)=(1)𝑛1𝑃𝑛𝑃(1),𝑛=𝐹𝑛(2)=(1)𝑛1𝐹𝑛𝑃(2),𝑛=𝑖2𝑛1Φ𝑛2=𝑖2𝑖2𝑛1Φ𝑛2,𝑃2𝑖𝑛=2𝑛+1𝑖4𝑎𝑛1𝐷𝑛12𝑎𝑖,𝑎=2𝑛+1𝑖4𝑎𝑛1𝐷𝑛12.𝑎𝑖,𝑎(3.1)

Example 3.2. If 𝑝=1 and 𝑞=2, then 𝑎𝑛 defined by (1.1) with initial conditions 𝑎0=0 and 𝑎1=1 are the Jacobsthal numbers 𝐽𝑛 (see Bergum et al. [19]). Thus Corollary 2.2 gives the expressions of 𝐽𝑛 as follows: 𝐽𝑛=2𝑖𝑛1𝑈𝑛1𝑖22=2𝑖𝑛1𝑈𝑛1𝑖22,𝐽𝑛=2𝑛1𝑃𝑛122=2𝑛1𝑃𝑛122,𝐽𝑛=2𝑛1𝐹𝑛12=2𝑛1𝐹𝑛12,𝐽𝑛=𝑖𝑛1Φ𝑛(𝑝i)=(𝑖)𝑛1Φ𝑛𝐽(𝑝𝑖),𝑛=2𝑛+1𝑖2𝑎𝑛1𝐷𝑛1𝑝𝑎𝑖2,𝑎=2𝑛+1𝑖2𝑎𝑛1𝐷𝑛1𝑝𝑎𝑖2.,𝑎(3.2)

Example 3.3. If 𝑝=3 and 𝑞=2, then 𝑎𝑛 defined by (1.1) with initial conditions 𝑎0=0 and 𝑎1=1 are the Mersenne numbers 𝑀𝑛=2𝑛1. From Corollary 2.2, we have 𝑀𝑛=2𝑛1𝑈𝑛1322=2𝑛1𝑈𝑛1322,𝑀𝑛=2𝑖𝑛1𝑃𝑛3𝑖22=2𝑖𝑛1𝑃𝑛3𝑖22,𝑀𝑛=2𝑖𝑛1𝐹𝑛3𝑖2=2𝑖𝑛1𝐹𝑛3𝑖2,𝑀𝑛=Φ𝑛(3)=(1)𝑛1Φ𝑛𝑀(3),𝑛=2𝑛+112𝑎𝑛1𝐷𝑛13𝑎2,𝑎=2𝑛+112𝑎𝑛1𝐷𝑛13𝑎2.,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 𝑎𝑛 defined by (1.1) with initial conditions 𝑎0=2 and 𝑎1=1 are the Lucas numbers 𝐿𝑛. Thus, besides (2.18), we have 𝐿𝑛=2𝑖𝑛𝑈𝑛𝑖2𝑖𝑛1𝑈𝑛1𝑖2=2(𝑖)𝑛𝑈𝑛𝑖2(𝑖)𝑛1𝑈𝑛1𝑖2,𝐿𝑛=2𝑃𝑛+112𝑃𝑛12=2(1)𝑛𝑃𝑛+112(1)𝑛1𝑃𝑛12,𝐿𝑛=2𝐹𝑛+1(1)𝐹𝑛(1)=2(1)𝑛𝐹𝑛+1(1)(1)𝑛1𝐹𝑛𝐿(1),𝑛𝑖=22𝑛Φ𝑛+1𝑖2𝑖2𝑛1Φ𝑛𝑖2𝑖=22𝑛Φ𝑛+1𝑖2𝑖2𝑛1Φ𝑛,𝐿2𝑖𝑛=2𝑛+3𝑖4𝑎𝑛𝐷𝑛𝑎𝑖,𝑎2𝑛+1𝑖4𝑎𝑛1𝐷𝑛1𝑎𝑖,𝑎=2𝑛+3𝑖4𝑎𝑛𝐷𝑛𝑎𝑖,𝑎2𝑛+1𝑖4𝑎𝑛1𝐷𝑛1.𝑎𝑖,𝑎(3.4)

Example 3.5. If 𝑝=2 and 𝑞=1, then 𝑎𝑛 defined by (1.1) with initial conditions 𝑎0=2 and 𝑎1=2 are the Pell-Lucas numbers 𝐴𝑛 (see Example 2 in [11]). Thus, from Corollary 2.4, we obtain 𝐴𝑛=2𝑖𝑛𝑇𝑛(𝑖)=2(𝑖)𝑛𝑇𝑛𝐴(𝑖),𝑛=2𝑖𝑛𝑈𝑛(𝑖)2𝑖𝑛1𝑈𝑛1(𝑖)=2𝑖𝑛𝑈𝑛(𝑖)2𝑖𝑛1𝑈𝑛1(𝐴𝑖),𝑛=2𝑃𝑛+1(1)2𝑃𝑛(1)=2(1)𝑛𝑃𝑛+1(1)𝑝(1)𝑛1𝑃𝑛(𝐴1),𝑛=2𝐹𝑛+1(2)2𝐹𝑛(2)=2(1)𝑛𝐹𝑛+1(2)𝑝(1)𝑛1𝐹𝑛(𝐴2),𝑛𝑖=22𝑛Φ𝑛+12𝑖2𝑖22𝑛1Φ𝑛2𝑖2𝑖=22𝑛Φ𝑛+12𝑖2𝑖22𝑛1Φ𝑛2,𝐴2𝑖𝑛=2𝑛+3𝑖4𝑎𝑛𝐷𝑛2𝑎𝑖,𝑎2𝑛+2𝑖4𝑎𝑛1𝐷𝑛12𝑎𝑖,𝑎=2𝑛+3𝑖4𝑎𝑛𝐷𝑛2𝑎𝑖,𝑎2𝑛+2𝑖4𝑎𝑛1𝐷𝑛12.𝑎𝑖,𝑎(3.5)

Example 3.6. If 𝑝=1 and 𝑞=2, then 𝑎𝑛 defined by (1.1) with initial conditions 𝑎0=2 and 𝑎1=1 are the Jacobsthal-Lucas numbers 𝐵𝑛 (see Example 2 in [11]). Thus, 𝐵𝑛=22𝑖𝑛𝑇𝑛𝑖22=22𝑖𝑛𝑇𝑛𝑖22,𝐵𝑛=22𝑖𝑛𝑈𝑛𝑖222𝑖𝑛1𝑈𝑛1𝑖22=22𝑖𝑛𝑈𝑛𝑖222𝑖𝑛1𝑈𝑛1𝑖22,𝐵𝑛=22𝑛𝑃𝑛+11222𝑛1𝑃𝑛122=22𝑛𝑃𝑛+11222𝑛1𝑃𝑛122,𝐵𝑛=22𝑛𝐹𝑛+1122𝑛1𝐹𝑛12=22𝑛𝐹𝑛+1122𝑛1𝐹𝑛12,𝐵𝑛=2𝑖𝑛Φ𝑛+1(𝑖)𝑖𝑛1Φ𝑛(𝑖)=2(𝑖)𝑛Φ𝑛+1(𝑖)(𝑖)𝑛1Φ𝑛𝐵(𝑖),𝑛=2𝑛+3𝑖2𝑎𝑛𝐷𝑛𝑎𝑖2,𝑎2𝑛+1𝑖2𝑎𝑛1𝐷𝑛1𝑎𝑖2,𝑎=2𝑛+3𝑖2𝑎𝑛𝐷𝑛𝑎𝑖2,𝑎2𝑛+1𝑖2𝑎𝑛1𝐷𝑛1𝑎𝑖2.,𝑎(3.6)

Example 3.7. If 𝑝=3 and 𝑞=2, then 𝑎𝑛 defined by (1.1) with initial conditions 𝑎0=2 and 𝑎1=3 are the Fermat numbers 𝑓𝑛 (see [20]). Thus, from Corollary 2.4, we obtain 𝑓𝑛=22𝑛𝑇𝑛322=22𝑛𝑇𝑛322,𝑓𝑛=22𝑛𝑈𝑛32232𝑛1𝑈𝑛1322=22𝑛𝑈𝑛32232𝑛1𝑈𝑛1322,𝑓𝑛=22𝑖𝑛𝑃𝑛+13𝑖2232𝑖𝑛1𝑃𝑛3𝑖22=22𝑖𝑛𝑃𝑛+13𝑖2232𝑖𝑛1𝑃𝑛3𝑖22,𝑓𝑛=22𝑖𝑛𝐹𝑛+13𝑖232𝑖𝑛1𝐹𝑛3𝑖2=22𝑖𝑛𝐹𝑛+13𝑖232𝑖𝑛1𝐹𝑛3𝑖2,𝑓𝑛=2Φ𝑛+1(3)3Φ𝑛(3)=2(1)𝑛Φ𝑛+1(3)3(1)𝑛1Φ𝑛𝑓(3),𝑛=2𝑛+312𝑎𝑛𝐷𝑛3𝑎2,𝑎(3)2𝑛+112𝑎𝑛1𝐷𝑛13𝑎2,𝑎=2𝑛+312𝑎𝑛𝐷𝑛3𝑎2,𝑎(3)2𝑛+112𝑎𝑛1𝐷𝑛13𝑎2.,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: 𝑃𝑛1,𝑦,𝐶(𝑥)=𝛼(𝑥)𝑃1,𝑦,𝐶𝑛1(𝑥)+𝐶2(2𝑥𝛼(𝑥)𝐶)(𝛽(𝑥))𝑛1,(3.8) where 𝑃𝑛1,𝑦,𝐶(𝑥) satisfies the recurrence relation of order 2, 𝑃𝑛1,𝑦,𝐶=𝑝𝑃1,𝑦,𝐶𝑛1+𝑞𝑃1,𝑦,𝐶𝑛2 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]), 1𝛼=𝐶𝑥+𝑥2,1𝐶𝑦𝛽=𝐶𝑥𝑥2.𝐶𝑦(3.9) For 𝑦=1 and 𝐶=1, we have 𝑃𝑛1,1,1(𝑥)=𝐹𝑛+1(2𝑥), where 𝐹𝑛(𝑥) are the Fibonacci polynomials, and we can write (3.8) as 𝐹𝑛+1(2𝑥)=𝛼(𝑥)𝐹𝑛(2𝑥)+(2𝑥𝛼(𝑥))(𝛽(𝑥))𝑛1=𝛼(𝑥)𝐹𝑛(2𝑥)+(𝛽(𝑥))𝑛,(3.10) where 𝛼(𝑥)=𝑥+𝑥2+1 and 𝛽(𝑥)=𝑥𝑥2+1. If 𝑥=1/2, then 𝐹𝑛(1)=𝐹𝑛, the Fibonacci numbers, and 𝛼12=1+521,𝛽2=152.(3.11)

Thus (3.10) yields the identity 𝐹𝑛+1=1+52𝐹𝑛+152𝑛,(3.12) or equivalently, 152𝐹𝑛+1+𝐹𝑛=152𝑛+1.(3.13)

Similarly, if 𝑥=1, then 𝐹𝑛(2)=𝑃𝑛, the Pell numbers, and 𝛼(1)=1+2,𝛽(1)=12.(3.14) Thus (3.10) yields the identity𝑃𝑛+1=1+2𝑃𝑛+12𝑛,(3.15) or equivalently, 12𝑃𝑛+1+𝑃𝑛=12𝑛+1.(3.16)

Substituting 𝑥=1/(22) into (3.10) and noting 𝐹𝑛(1/2)=𝐽𝑛/(2)𝑛, where 𝐽𝑛 are the Jacobsthal numbers, we obtain the identity 𝐽𝑛+12𝐽𝑛=(1)𝑛.(3.17) When 𝑥=3𝑖/(22), 𝐹𝑛(3𝑖/(22))=𝑀𝑛/(2𝑖)𝑛1, the Mersenne numbers. Hence (3.10) gives 𝑀𝑛+1𝑀𝑛=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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