The main purpose of this paper is using the combinatorial method, the properties of the power series and characteristic roots to study the computational problem of the symmetric sums of a certain second-order linear recurrence sequences, and obtain some new and interesting identities. These results not only improve on some of the existing results, but are also simpler and more beautiful. Of course, these identities profoundly reveal the regularity of the second-order linear recursive sequence, which can greatly facilitate the calculation of the symmetric sums of the sequences in practice.
the second-order linear recurrence sequence; convolution sums; new identity; recurrence formula
The defined of second-order linear recurrence sequence is
where n is integers with .
For convenience, we also extend the recursive property of to all negative integers.
We taking , , with , in (1), then becomes the famous Fibonacci polynomial sequence . That is,
Especially when , becomes known as the Fibonacci sequence.
Let and denote the two roots of the characteristic equation . Then we have
where denotes the Lucas polynomials, and denotes the Lucas sequence.
If we take , in (1), then is Chebyshov polynomials of the second kind with and . Chebyshov polynomials of the first kind is defined by for all with and . Let , are two characteristic roots of the polynomial , then (see )
Many scholars have studied , and obtained a series of valuable research results. For example, Yi Yuan and Zhang Wenpeng  proved the following conclusion: For any positive integer n and k, one has the identity
where denotes the summation is taken over all k-dimension nonnegative integer coordinates such that .
Ma Yuankui and Zhang Wenpeng  also studied this problem, and proved the following result:
where is defined by , , and
for all positive integers .
On the other hand, Zhang Yixue and Chen Zhuoyu  studied the properties of Chebyshov polynomials, and proved the following identity:
where is a second order non-linear recurrence sequence defined by , , and for all .
Many other papers related to Fibonacci numbers, Fibonacci polynomials, Chebyshov polynomials and second-order linear recurrence sequences can also be found in references [5,6,7,8,9,10,11,12,13,14,15,16,17,18], here we will no longer list them one by one.
After careful analysis of the research content in [1,2,3,4], we think it can be summarized as a sentence: That is, to study the symmetry sum problem of the generalized second-order linear recursive sequence. Of course, they are meaningful to study these problems. It not only reveals the profound properties of the generalized second-order linear recursive polynomials and sequences, but also greatly simplifies the calculation of the symmetry sums of these polynomials and sequences in practice.
Inspired by [1,2,3,4], in this paper, we will use a new method to study the computational problem of the symmetry sums of a certain second-order linear recurrence sequences, and give a simple and beautiful generalized conclusion. That is, we will use the elementary methods and the symmetry properties of the characteristic roots to prove the following results:
Let denotes any second-order linear recurrence sequence with and . Then we have the identity
It is clear that if we taking and , then from Theorem 1 we may immediately deduce the following:
For any positive integers n and k, we have the identity
For any positive integer m, n and k, we have the identity
It is clear that our Corollary 1 and Corollary 2 are much easier than the results in [1,2,3,4]. If with and and with and are two different second-order linear recurrence sequences, such that the polynomials and co-prime. That is, . Then we define sequence as follows:
For the sequence defined in (2), we have the following conclusion:
The sequence is a fourth-order linear recurrence sequence, and it satisfy the fourth-order linear recurrence formula
where , , and
Taking , , and , from our Theorem 2 we can deduce the following result:
For any integer , we define the polynomials sequence
Then is a fourth-order linear recurrence polynomials, and it satisfy the recurrence formula
for all integers , where , , and , and denote the Fibonacci polynomials and Chebyshov polynomials of the second kind respectively.
2. Proof of the Theorem
In this section, we will prove our main results directly. First we prove Theorem 1.
Proof of Theorem1.
It is clear that the characteristic equation of the sequence is . Let and are the two characteristic roots of the equation . Then we have
The generating function of the sequence is
where and .
For any positive integer k, we have the identity
On the other hand, from the properties of the power series we have
Thus, from (5) and the properties of the power series we have
Combining (4), (6) and note that and the symmetry of and we can deduce the identity
From the definitions A and B we have
So for any integer r, from the definition of we have
Now combining (7) and (8) we may immediately deduce the identity
This proves Theorem 1. □
Proof of Theorem2.
Note that , so from the definitions of , , and we have
where , , and are different each others.
It is clear that from the definitions sequences and we have
On the other hand, from the definition and properties of the fourth-order linear recurrence sequence we also have
where , , ,
From (11) and (12) we know that the sequence
is a fourth-order recurrence sequence, and it satisfy the recurrence Formula (12).
This completes the proof of Theorem 2. □
Writing—original draft: Y.L.; Writing—review and editing: X.L.
This work is supported by the Xizang N. S. F. (XZ2017ZRG-65) and the N. S. F. (11771351) and (11826205) of China.
The authors would like to thank the referees for their very helpful and detailed comments, which have significantly improved the presentation of this paper.
Conflicts of Interest
The authors declare no conflict of interest.
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