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
Tian-Xiao He,1Peter J.-S. Shiue,2 and Tsui-Wei Weng3
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
Academic Editor: W. Liu
Received06 Jul 2011
Accepted25 Aug 2011
Published26 Oct 2011
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.
Many number and polynomial sequences can be defined, characterized, evaluated, and/or classified by linear recurrence relations with certain orders. A number sequence is called sequence of order 2 if it satisfies the linear recurrence relation of order 2:
for some nonzero constants and and initial conditions and . In Mansour , the sequence defined by (1.1) is called Horadamβs sequence, which was introduced in 1965 by Horadam . In  also the generating functions for powers of Horadamβs 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 , Hsu , Strang , Wilf , etc.) In , Benjamin and Quinn presented many elegant combinatorial meanings of the sequence defined by recurrence relation (1.1). For instance, counts the number of ways to tile an -board (i.e., board of length ) with squares (representing 1s) and dominoes (representing 2s) where each tile, except the initial one, has a color. In addition, there are colors for squares and colors for dominoes. In particular, Aharonov et al. (see ) have proved that the solution of any sequence of numbers that satisfies a recurrence relation of order 2 with constant coefficients and initial conditions and , called the primary solution, can be expressed in terms of the Chebyshev polynomial values. For instance, the authors show and , where and , respectively, are the Fibonacci numbers and Lucas numbers, and and are the Chebyshev polynomials of the first kind and the second kind, respectively. Some identities drawn from those relations were given by Beardon in . Marr and Vineyard in  use the relationship to establish explicit expression of five-diagonal Toeplitz determinants. In , the first two authors presented a new method to construct an explicit formula of generated by (1.1). For the sake of the reader's convenience, we cite this result as follows.
Proposition 1.1 (see ). Let be a sequence of order 2 satisfying linear recurrence relation (1.1), and let and be two roots of of quadratic equation . Then
A sequence of the generalized Gegenbauer-Humbert polynomials is defined by the expansion (see, e.g., Comtet , Gould , Lidl et al. , the two authors with He et al. )
where , and are real numbers. As special cases of (1.3), we consider as follows (see ): , the Chebyshev polynomial of the second kind,, the Legendre polynomial,, the Pell polynomial,, the Fibonacci polynomial,, the Fermat polynomial of the first kind,, the Dickson polynomial of the second kind, (see, e.g., ),
where is a real parameter, and is the Fibonacci number. In particular, if , the corresponding polynomials are called the Gegenbauer polynomials (see ). More results on the Gegenbauer-Humbert-type polynomials can be found in  by Hsu and in  by the second author and Hsu, and so forth.
Similarly, for a class of the generalized Gegenbauer-Humbert polynomial sequences defined by
for all with initial conditions
the following theorem has been obtained in .
Theorem 1.2 (see ). Let . The generalized Gegenbauer-Humbert polynomials defined by expansion (1.3) can be expressed as
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 and , 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 , the first formula in (1.2) can be written as
where the last step is due to and being solutions of . Noting that and , we may further write the above last expression of as
Denote and . Comparing expressions (2.2) and (1.6), we have reason to consider the following transform: for a nonzero real or complex number , we set
for a certain depending on , , and , which we will find out later. Denote and ; that is, and are roots of . By adding the two equations in (2.3) side by side, we obtain . Thus, when , the equations in (2.2) hold. Meanwhile, by using , we have
where . Therefore, we obtain
We first consider the case of .
We now substitute , , , and into (2.2) and simplify as follows:
Similarly, for , we have
Therefore, we obtain our main result.
Theorem 2.1. Let sequence be defined by () with initial conditions and . Then, can be presented as (2.7) and (2.8). In particular, for , and , respectively, one has
where , , , , and are the th 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 and , we have the following corollaries.
Corollary 2.2. Let sequence be defined by () with initial conditions and . Then
Corollary 2.3. Let sequence be defined by () with initial conditions and . Then
If , then Corollary 2.2 gives the primary solutions of recurrence relation (1.1) in terms of the th 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 , then are the Fibonacci numbers . Thus,
where was shown in  and was given by Chen and Louck in . From the above expressions of , we may obtain many identities. For instance, we have
and so forth.
We now give another special case of Theorem 2.1 for the sequence defined by (1.1) with initial cases and .
Corollary 2.4. Let sequence be defined by () with initial conditions and . Then
In addition, one has
where 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 , one easily sees
From the basic relation between Chebyshev polynomials of the first and the second kinds (see, e.g., (1.7) in  by Mason and Handscomb), , the last expression of implies (2.15). Equation (2.16) can be proved similarly.
As an example, the Lucas number sequence defined by (1.1) with and initial conditions and has the explicit formula for its general term:
3. Examples and Applications
We first give some examples of Corollary 2.2 for sequences that are primary solutions of (1.1).
Example 3.1. If and , then defined by (1.1) with initial conditions and are the Pell numbers . Thus, from Corollary 2.2, we have
Example 3.2. If and , then defined by (1.1) with initial conditions and are the Jacobsthal numbers (see Bergum et al. ). Thus Corollary 2.2 gives the expressions of as follows:
Example 3.3. If and , then defined by (1.1) with initial conditions and are the Mersenne numbers . From Corollary 2.2, we have
Next, we give several examples of nonprimary solutions of (1.1) by using Corollary 2.4.
Example 3.4. If and , then defined by (1.1) with initial conditions and are the Lucas numbers . Thus, besides (2.18), we have
Example 3.5. If and , then defined by (1.1) with initial conditions and are the Pell-Lucas numbers (see Example 2 in ). Thus, from Corollary 2.4, we obtain
Example 3.6. If and , then defined by (1.1) with initial conditions and are the Jacobsthal-Lucas numbers (see Example 2 in ). Thus,
Example 3.7. If and , then defined by (1.1) with initial conditions and are the Fermat numbers (see ). Thus, from Corollary 2.4, we obtain
Using the relationship established above, we may obtain some identities of number sequences and polynomial value sequences. Theorem 3.2 in  presented a generalized Gegenbauer-Humbert polynomial sequence identity:
where satisfies the recurrence relation of order 2, with coefficients and , and and . Clearly (see (19) and (20) in ),
For and , we have , where are the Fibonacci polynomials, and we can write (3.8) as
where and . If , then , the Fibonacci numbers, and
Similarly, if , then , the Pell numbers, and
Thus (3.10) yields the identity
Substituting into (3.10) and noting , where are the Jacobsthal numbers, we obtain the identity
When , , the Mersenne numbers. Hence (3.10) gives .
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.
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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