On Sequences of Numbers and Polynomials Defined by Linear Recurrence Relations of Order 2
Here we present a new method to construct the explicit formula of a sequence of numbers and polynomials generated by a linear recurrence relation of order 2. The applications of the method to the Fibonacci and Lucas numbers, Chebyshev polynomials, the generalized Gegenbauer-Humbert polynomials are also discussed. The derived idea provides a general method to construct identities of number or polynomial sequences defined by linear recurrence relations. The applications using the method to solve some algebraic and ordinary differential equations are presented.
Many number and polynomial sequences can be defined, characterized, evaluated, and classified by linear recurrence relations with certain orders. A number sequence is called sequence of order if it satisfies the linear recurrence relation of order :
for some nonzero constants and and initial conditions and . In Mansour , the sequence defined by (1.1) is called Horadam’s sequence, which was introduced in 1965 by Horadam . The work in  also obtained the generating functions for powers of Horadam’s sequence. 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 n-board (i.e., board of length ) with squares (representing s) and dominoes (representing s) where each tile, except the initial one has a color. In addition, there are colors for squares and colors for dominoes. In this paper, we will present a new method to construct an explicit formula of generated by (1.1). The key idea of our method is to reduce the relation (1.1) of order to a linear recurrence relation of order :
for some constants and and initial condition via geometric sequence. Then, the expression of the general term of the sequence of order can be obtained from the formula of the general term of the sequence of order :
The method and some related results on the generalized Gegenbauer-Humbert polynomial sequence of order as well as a few examples will be given in Section 2. Section 3 will discuss the application of the method to the construction of the identities of sequences of order . There is an extension of the above results to higher order cases. In Section 4, we will discuss the applications of the method to the solution of algebraic equations and initial value problems of second-order ordinary differential equations.
2. Main Results and Examples
Let and be two roots of of quadratic equation We may write (1.1) as
where and satisfy and . Therefore, from (2.1), we have
which implies that is a geometric sequence with common ratio . Hence,
Let . We may write (2.4) as
If , by using (1.3), we immediately obtain
Similarly, if , then (1.3) implies
We may summarize the above result as follows.
Proposition 2.1. Let be a sequence of order satisfying linear recurrence relation (2.1). Then
In particular, if satisfies the linear recurrence relation (1.1) with , namely,
then the equation has two solutions:
From Proposition 2.1, we have the following corollary.
Corollary 2.2. Let be a sequence of order satisfying the linear recurrence relation . Then
where is defined by (2.11).
Similarly, let be a sequence of order satisfying the linear recurrence relation . Then
where and are solutions of the equation .
The first special case (2.12) was studied by Falbo in . If , the sequence is clearly the Fibonacci sequence. If (), the corresponding sequence is the sequence of numerators (when two initial conditions are and ) or denominators (when two initial conditions are and ) of the convergent of a continued fraction to : , , , called the closest rational approximation sequence to . The second special case is also a corollary of Proposition 2.1. If (), is the Jacobsthal sequence (see Bergum et al. ).
Remark 2.3. Proposition 2.1 can be extended to the linear recurrence relations of order 2 with more general form: for . It can be seen that the above recurrence relation is equivalent to the form (1.1) , where and
We now show some examples of the applications of our method including the presentation of much easier proofs of some well-known formulas of the sequences of order .
Remark 2.4. Denote We may write relation and into a matrix form with respect to matrix defined above. Thus . To find explicit expression of , the real problem is to calculate . The key lies in the eigenvalues and eigenvectors. The eigenvalues of are precisely and , which are two roots of the characteristic equation for the matrix . However, an obvious identity can be obtained from by taking determinants on the both sides: (see, e.g.,  for more details).
Example 2.5. Let be the Fibonacci sequence with the linear recurrence relation , where and are assumed to be and , respectively. Thus, the recurrence relation is a special case of (1.1) with and the special case of the sequence in Corollary 2.2, which can be written as (2.1) with Since , from (2.12) we have the expression of as follows:
Example 2.6. We have mentioned above that the denominators of the closest rational approximation to form a sequence satisfying the recurrence relation . With an additional initial condition , the sequence becomes the Pell number sequence: , which also satisfies the recurrence relation . Using formula (2.23) in Corollary 2.2, we obtain the general term of the Pell number sequence:
The numerators of the closest rational approximation to are half the companion Pell numbers or Pell-Lucas numbers. By adding in initial condition 2, we obtain the Pell-Lucas number sequence , which satisfies . Similarly, Corollary 2.2 gives
We now consider the sequence of the sums of Pell number: , which satisfies the recurrence relation
From Remark 2.3, the above expression can be transfered to an equivalent form
where . Using Corollary 2.2, one easily obtain
If the coefficients of the linear recurrence relation of a function sequence of order are real or complex-value functions of variable , that is,
we obtain a function sequence of order with initial conditions and . In particular, if all of , , and are polynomials, then the corresponding sequence is a polynomial sequence of order . Denote the solutions of by and . Then Similar to Proposition 2.1, we have
Example 2.8. Consider the Chebyshev polynomials of the first kind, , defined by which satisfies the recurrence relation with and . Thus the corresponding , , and are, respectively, , , , and , which yields , , and . Substituting the quantities into (2.7) yields All the Chebyshev polynomials of the second kind, third kind, and fourth kind satisfy the same recurrence relationship as the Chebyshev polynomials of the first kind with the same constant initial term . However, they possess different linear initial terms, which are , , and , respectively (see, e.g., Mason and Handscomb  and Rivlin ). We will give the expression of the Chebyshev polynomials of the second kind later by sorting them into the class of the generalized Gegenbauer-Humbert polynomials. As for the Chebyshev polynomials of the third kind, , and the Chebyshev polynomials of the fourth kind, , when , we clearly have the following expressions using a similar argument presented for the Chebyshev polynomials of the first kind:
Example 2.9. In , André-Jeannin studied the generalized Fibonacci and Lucas polynomials defined, respectively, by
where and are real parameters. Clearly, . Using Proposition 2.7, we obtain
From the last expression, we also see
A sequence of the generalized Gegenbauer-Humbert polynomials is defined by the expansion (see, e.g., Gould  and He et al. ):
where , and are real numbers. As special cases of (2.33), we consider as follows (see ): where is a real parameter and is the Fibonacci number.
Theorem 2.10. Let . The generalized Gegenbauer-Humbert polynomials defined by expansion (2.33) can be expressed as
Proof. Taking derivative with respect to to the two sides of (2.33) yields
Then, substituting the expansion of of (2.33) into the left-hand side of (2.36) and comparing the coefficients of term on both sides, we obtain
By transferring , we have
for all with
Thus, if , satisfies linear recurrence relation
Therefore, we solve , where and , for and obtain solutions:
where . Hence, Proposition 2.7 gives the formula of as where and are shown as (2.41). This completes the proof.
Remark 2.11. We may use recurrence relation (2.40) to define various polynomials that were defined using different techniques. Comparing recurrence relation (2.40) with the relations of the generalized Fibonacci and Lucas polynomials shown in Example 2.9, with the assumption of and , we immediately know that
defines the Chebyshev polynomials of the second kind,
defines the Pell polynomials, and
defines the Fibonacci polynomials.
In addition, in , Lidl et al. defined the Dickson polynomials are also the special case of the generalized Gegenbauer-Humbert polynomials, which can be defined uniformly using recurrence relation (2.40), namely,
with and . Thus, the general terms of all of above polynomials can be expressed using (2.35).
Example 2.12. For , using (2.35), we obtain the expression of the Chebyshev polynomials of the second kind:
where . Thus, .
For and , formula (2.35) gives the expression of a Pell polynomial of degree :
Similarly, let and , the Fibonacci polynomials are
and the Fibonacci numbers are which has been presented in Example 2.5.
Finally, for and , we have Fermat polynomials of the first kind:
where . From the expressions of Chebyshev polynomials of the second kind, Pell polynomials, and Fermat polynomials of the first kind, we may get a class of the generalized Gegenbauer-Humbert polynomials with respect to defined as follows.
Definition 2.13. The generalized Gegenbauer-Humbert polynomials with respect to , denoted by , are defined by the expansion by or equivalently, by with and , where . In particular, , and are, respectively, Pell polynomials, Chebyshev polynomials of the second kind, and Fermat polynomials of the first kind.
3. Identities Constructed from Recurrence Relations
From (2.2) we have the following result.
Proposition 3.1. A sequence of order satisfies linear recurrence relation (2.1) if and only if it satisfies the nonhomogeneous linear recurrence relation of order 1 with the form where is uniquely determined. In particular, if , then is equivalent to , where .
Proof. The necessity is clearly from (2.1). We now prove sufficiency. If sequence satisfies the nonhomogeneous recurrence relation of order shown in (3.1), then by substituting into the above equation, we obtain . Thus, (3.1) can be written as which implies that satisfies the linear recurrence relation of order : with and . In particular, if , then and , which yields the special case of the proposition.
An obvious example of the special case of Proposition 3.1 is the Mersenne number (), which satisfies the linear recurrence relation of order : (with and ) and the nonhomogeneous recurrence relation of order : (with ). It is easy to check that sequence satisfies both the homogeneous recurrence relation of order 2, , and the nonhomogeneous recurrence relation of order , , where and .
where the last step is due to . Therefore, we give a simple identity
which is shown in [16, () page 122] by Koshy. Similarly, we have
where the last step is due to . The above identity can be written as
The same argument yields or equivalently,
Let be the Lucas number sequence with and , which satisfies recurrence relation (2.2) with the same and for the Fibonacci number sequence. Then, using the same argument, we have
(see [16, page 129]).
We now extend the above results regarding Fibonacci and Lucas numbers to more general sequences presented by Niven et al. in . Let and be two sequences defined, respectively, by the linear recurrence relations of order :
with initial conditions and and and , respectively. Clearly, if , then and are, respectively, Fibonacci and Lucas numbers. From (3.2), we immediately have
Multiplying to both sides of the above equation yields
Similarly, we obtain
Using (3.2) we can also obtain the identity
which implies (3.10) when .
Aharonov et al. (see ) have proved that the solution of any sequence of numbers that satisfies a recurrence relation of order with constant coefficients and initial conditions and can be expressed in terms of Chebyshev polynomials. For instance, the authors show and . Thus, we have identities
In , Chen and Louck obtained . Thus we have identity
where the last step is from (3.6) with transform . Thus, we have . If is odd, the left-hand side of (3.18) is If is even, the left-hand side of (3.18) becomes which completes the proof of Sofo’s identity.
Theorem 3.2. Let be a sequence of numbers or polynomials defined by the linear recurrence relation () with initial conditions and , and let and . Then we have identity In particularly, if , the sequence of the generalized Gegenbauer-Humbert polynomial is defined by (2.33), then we obtain the polynomial identity: where and are shown in (2.41). For (i.e., the generalized Gegenbauer-Humbert polynomials with respect to ), we denote and have
Actually, (3.22) can also be proved directly. Similarly, for the Chebyshev polynomials of the first kind , we have the identity
Let , the above identity becomes
which is equivalent to
It is clear that if is bounded and , from (3.20) we have
The method presented in this paper cannot be extended to the higher-order setting. However, we may use the idea and a similar argument to derive some identities of sequences of order greater than . For instance, for a sequence of numbers or polynomials that satisfies the linear recurrence relation of order 3:
we set the equation
Using transform , we can change the equation to the standard form , which can be solved by Vieta’s substitution . The formulas for the three roots, denoted by , , are sometimes known as Cardano’s formula. Thus, we have
Denote . Then (3.28) can be written as
Therefore, from the identity , we obtain identity in terms of :
where can be found uniquely from and , that is,
Remark 3.3. Similar to the particular case shown in Proposition 3.1, we may find the equivalence between the nonhomogeneous recurrence relation of order , , and the homogeneous recurrence relation of order , , where
Example 3.4. As an example, we consider the tribonacci number sequence generated by (). Solving , we obtain
Substituting , , , and (with the assumption ) and into (3.33), we obtain an identity regarding the tribonacci number sequence :
where and .
For the sequence defined by
with initial conditions and . Then, the first few numbers of the sequence are . The three roots of are , , and . Therefore, by assuming , we obtain the corresponding and and the following identity for the above-defined sequence: for all . From  by Haye, (), where are Stirling numbers of the second kind. Hence, we obtain an identity of the Stirling numbers of the second kind:
The idea to reduce a linear recurrence relation of order to order can be extended to the higher order cases. In general, if we have a sequence satisfying the linear recurrence relation of order :
Assume the equation has solutions (). Denote . Then the above recurrence relation can be reduced to a linear recurrence relation of order for sequence . Using this process, we may obtain the explicit formula of and/or identities in terms of if we know the solution of the last equation and/or the identities in terms of sequence .
The process shown in Proposition 3.1 can be applied conversely to elevate a nonhomogenous recurrence relation of order to a homogeneous recurrence relation of order .
4. Solutions of Algebraic Equations and Differential Equations
The results presented in Sections 2 and 3 have more applications. In this section, we will discuss the applications in the solutions of algebraic equations and initial value problems of second-order ordinary differential equations.
First, we consider roots of polynomials or the solution of , where is the Fibonacci sequence. Using the identity we immediately know that the largest root of is
Indeed, only changes its coefficient signs once, which implies that it has only one positive root and all of its other roots must be negative, for example, . In , Wall proved the largest root of is using a more complicated manner.
We may write the identity as
where is the th Lucas number. Similarly, we have
Multiplying the last two equations side by side yields or equivalently, The last expression means is a perfect square, or equivalently, if is a Fibonacci number, then is a perfect square. This result is a part of Gessel’s results in , but the method we used seems simpler. In addition, the above result also shows that the Pell’s equation has a solution .
The above results on the Fibonacci sequence can be extended to the sequence shown in  and Section 3. Consider polynomial with . We can see the largest root of is , which implies Wall’s result when . In addition, because of
Hence, is a perfect square, which implies the special case for the Fibonacci sequence that has been presented. Therefore, Pell’s equation () has a solution .
We now use the method presented in Section 2 to reduce an initial problem of a second order ordinary differential equation:
of second-order ordinary differential equation with constant coefficients to the problem of linear equations. Let , and let and be solution(s) of . Denote
Then , and the original initial problem of the second order is split into two problems of first order by using the method shown in (2.2) for :
Thus, we obtain the solutions
The above technique can be extended to the initial problems of higher-order ordinary differential equations. In this paper, we presented an elementary method for construction of the explicit formula of the sequence defined by the linear recurrence relation of order and the related identities. Some other applications in solutions of algebraic and differential equations and some extensions to the higher dimensional setting are also discussed. However, besides those applications, more applications in combinatorics and the combinatorial explanations of our given formulas still remain much to be investigated.
Dedicated to Professor L. C. Hsu on occasion of his 90th birthday. The authors wish to thank the referees for their helpful comments and suggestions.
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