The power sum of balancing polynomials and their divisible properties

1.   Introduction

Behera and Panda ^[1] introduced the concept of balancing numbers ${{B}_{n}}$, a positive integer $n$ is a balancing number if

That $r$ is the balancer corresponding to the balancing number $n$. The balancing numbers ${{B}_{n}}$ satisfy the relation ${{B}_{n+1}} = 6{{B}_{n}}-{{B}_{n-1}}\text{ }, n\ge 1$ with ${{B}_{0}} = 0, \text{ }{{B}_{1}} = 1$. The sequence ${{C}_{n}} = \sqrt{8B_{n}^{2}+1}$ is called a Lucas-balancing number. The Lucas-balancing number satisfies same relation ${{C}_{n+1}} = 6{{C}_{n}}-{{C}_{n-1}}\text{ }, n\ge 1$ with ${{C}_{0}} = 1, \text{ }{{\text{C}}_{1}} = 3$. Some conclusions about these two sequences can be found in the references ^[2,3]. The balancing polynomial and the Lucas-balancing polynomial are natural extensions of balancing numbers and Lucas-balancing numbers.

For any integer $n\ge0$, the balancing polynomials ${{B}_{n}}\left(x \right)$ and Lucas-balancing polynomials ${{C}_{n}}\left(x \right)$ are defined as follows (see Frontczak and Goy ^[4]):

where $\left\lfloor x \right\rfloor$ denotes the greatest integer $\le x$.

${{B}_{n}}\left(x \right)$ and ${{C}_{n}}\left(x \right)$ are the second-order linear recurrence polynomials, they satisfy the recurrence formulae (see Frontczak and Goy ^[4]):

${{B}_{n+1}}\left(x \right) = 6x{{B}_{n}}\left(x \right)-{{B}_{n-1}}\left(x \right)$ for all $n\ge 1$, with ${{B}_{0}}\left(x \right) = 0$, ${{B}_{1}}\left(x \right) = 1$,

${{C}_{n+1}}\left(x \right) = 6x{{C}_{n}}\left(x \right)-{{C}_{n-1}}\left(x \right)$ for all $n\ge 1$, with ${{C}_{0}}\left(x \right) = 1$, ${{C}_{1}}\left(x \right) = 3x$.

The closed forms which are also called Binets formulas for balancing polynomials and Lucas-balancing polynomials are given by

where $\alpha \left(x \right) = 3x+\sqrt{9{{x}^{2}}-1}, \ \beta \left(x \right) = 3x-\sqrt{9{{x}^{2}}-1}.$ The relations ${{B}_{n}}\left(-x \right) = {{\left(-1 \right)}^{n+1}}{{B}_{n}}\left(x \right)$ and ${{C}_{n}}\left(-x \right) = {{\left(-1 \right)}^{n}}{{C}_{n}}\left(x \right)$ follow from $\alpha \left(-x \right) = -\beta \left(x \right)\text{ }$and$\text{ }-\alpha \left(x \right) = \beta \left(x \right).$ If we take $x = 1$, then $\left\{ {{B}_{n}}\left(x \right) \right\}$ becomes balancing sequences $\left\{ {{B}_{n}} \right\}$, and $\left\{ {{C}_{n}}\left(x \right) \right\}$ becomes Lucas-balancing sequences $\left\{ {{C}_{n}} \right\}$. Such balancing numbers and balancing polynomials have been widely studied in recent years. Frontczak ^[5] proves the sum of powers of balancing polynomials and Lucas balancing polynomials:

Kim and Kim ^[6] used nine orthogonal polynomials to represent the sum of the finite product of balancing polynomials to obtained the following result:

Ray ^[7] studied the divisible property of balancing numbers and Lucas-balancing number obtained the congruence:

For any integer $n\ge 0$, the famous Fibonacci polynomials ${{F}_{n}}\left(x \right)$ and Lucas polynomials ${{L}_{n}}\left(x \right)$ are defined as follows (see Wang and Zhang ^[8]) :

${{F}_{n+2}}\left(x \right) = x{{F}_{n+1}}\left(x \right)+{{F}_{n}}\left(x \right)$, with ${{F}_{0}}\left(x \right) = 0$, ${{F}_{1}}\left(x \right) = 1$,

${{L}_{n+2}}\left(x \right) = x{{L}_{n+1}}\left(x \right)+{{L}_{n}}\left(x \right)$, with ${{L}_{0}}\left(x \right) = 2$, ${{L}_{1}}\left(x \right) = x$.

These polynomials and sequences have some similarities in structure and properties. Kim ^{[9,10,11,12,13]} obtained many meaningful results by studying connections between polynomials. Mathematics has a wide range of applications in other disciplines, see ^[14,15,16]. We can obtain some divisible properties of polynomials and sequences in references ^[17,18,19]. For example, Wang and Zhang ^[8] proved the congruence of the sum of powers of Fibonacci numbers. That is

In this paper, we use the properties of balancing polynomials and Lucas balancing polynomials to study the divisible properties of $\sum\limits_{m = 0}^{h}{B_{{{2}^{s}}ml}^{2n+1}}\left(x \right)$ and $\sum\limits_{m = 0}^{h}{C_{{{2}^{s}}ml}^{2n+1}}\left(x \right)$ to get more general results. That is, we shall prove the following two theorems.

Theorem 1. Let $n$ and $h$ be non-negative integer with $h\ge 1$, $s$ and $l$ be positive integers. Then we have the congruence

Theorem 2. Let $n$ and $h$ be non-negative integers with $h\ge 1$, $s$ and $l$ be positive integers. Then we have the congruence

From the two theorems, we can obtain the following corollaries.

Corollary 1. For any non-negative integers $n$ and $h$ with $h\ge 1$, we have

Corollary 2. For any non-negative integers $n$ and $h$ with $h\ge 1$, we have

Corollary 3. For any non-negative integers $n$ and $h$ with $h\ge 1$, and $s$ and $l$ be positive integers, we have

Corollary 4. For any non-negative integers $n$ and $h$ with $h\ge 1$, and $s$ and $l$ be positive integers, we have

For Chebyshev polynomials of the first kind ${{T}_{n+1}}\left(x \right) = 2x{{T}_{n}}\left(x \right)-{{T}_{n-1}}\left(x \right)$ with ${{T}_{0}}\left(x \right) = 1$, ${{T}_{1}}\left(x \right) = x$ and Chebyshev polynomials of the second kind ${{U}_{n+1}}\left(x \right) = 2x{{U}_{n}}\left(x \right)-{{U}_{n-1}}\left(x \right)$ with ${{U}_{0}}\left(x \right) = 1$, ${{U}_{1}}\left(x \right) = 2x.$ The balancing polynomials possess a simple connection to Chebyshev polynomials of the first and second kind ${{T}_{n}}\left(x \right)$ and ${{U}_{n}}\left(x \right)$, specifically ${{B}_{n}}\left(x \right) = {{U}_{n-1}}\left(3x \right)$ and ${{C}_{n}}\left(x \right) = {{T}_{n}}\left(3x \right)$.

Taking $x = \frac{1}{3}x$ in Theorem 1, we can get the following,

Corollary 5. For any non-negative integers $n$ and $h$ with $h\ge 1$, and $s$ and $l$ be positive integers, we have

Taking $s = 1$ and $x = \frac{1}{3}x$ in Theorem 2, we can get the following,

Corollary 6. For any non-negative integers $n$ and $h$ with $h\ge l$, and $l$ be positive integers, we have

2.   Some lemmas

In the following, we use the properties of balancing polynomials and Lucas-balancing polynomials to prove our next several lemmas, which will help us better complete the proofs of the theorems.

Lemma 1. Let $s$ and $h$ be positive integers. Then, for any integers $n$ and $l$, we have the identity

Proof. We prove this polynomial congruence by complete induction for $n\ge 0$. It is clear that Lemma 1 is true for $n = 0$. If $n = 1$, then note ${{C}_{{{2}^{^{s-1}}}3l\left(2\text{h}+1 \right)}}\left(x \right) = 4C_{{{2}^{s-1}}l\left(2h+1 \right)}^{3}\left(x \right)-3{{C}_{{{2}^{s-1}}l\left(2h+1 \right)}}\left(x \right)$, we have

That is to say, Lemma 1 is true for $n = 1$.

Suppose that Lemma 1 is true for all positive integers $0\le n\le j$. That is,

for all $0\le n\le j$.

Then, for $n = j+1\ge 2$, we have

and

Applying inductive hypothesis (2.1), we have

That is to say, the Lemma 1 is true for $n = j+1$.

Now Lemma 1 follows from complete induction. □

Lemma 2. Let $s$ and $h$ be positive integers. Then, for any integers $n$ and $l$, we have the identity

Proof. We can also prove Lemma 2 by complete induction. If $n = 0,$ then it is clear that Lemma 2 is true. If $n = 1$, then note

we have

So Lemma 2 is true for $n = 1$. Suppose that Lemma 2 is true for positive integers $0\le n\le j$. That is,

for all $0\le n\le j$.

Then, for $n = k+1$, note the identities

and

applying inductive hypothesis (2.2), we have

This completes the proof of Lemma 2. □

Lemma 3. For all non-negative integers $r$ and real numbers $w, v,$ we have the identity

in which $\left[x \right]$ denotes the greatest integer $\le x$.

Proof. The formula due to E. Waring ^[20] and can be found in H. W. Gould ^[21]. □

3.   Proofs of the theorems

We shall prove our theorems by mathematical induction. Taking $w = {{\alpha }^{{{2}^{s}}ml}}\left(x \right)$, $v = -{{\beta }^{{{2}^{s}}ml}}\left(x \right)$ and $r = 2n+1$ in Lemma 3, we notice that $wv = -1$, from the expression of ${{B}_{n}}\left(x \right)$ we have

For any integer $h\ge 1$, from (3.1) we have

Note the identities

And

Combining (3.2) and (3.3) we have

Now we apply (3.5) and mathematical induction to prove Theorem 1. If $n = 1$, then from (3.5) we have

From Lemma 1 we know that

Combining (3.6) and (3.7) we know that Theorem 1 is true for $n = 1$.

Suppose that Theorem 1 is true for all integers $1\le n\le j$. Then, for $n = j+1$, from (3.5) we have

From Lemma 1 we have

Applying inductive hypothesis (3.8), we have

Combining (3.7)–(3.10) and Lemma 1, we have the congruence

This completes the proof of Theorem 1 by mathematical induction.

Now we prove Theorem 2, we have

Applying (3.11), Lemma 2 and the method of proving Theorem 1, we can deduce the congruence

□

4.   Conclusions

In this paper, we study the divisible property of the general power sum of balancing polynomials and Lucas-balancing polynomials. By taking specific values for $s$ and $l$ in the Theorems 1 and 2, similar results can be obtained as studied in the literature. In this paper, we take $x = 1$ and obtain the divisible property of the sequence $\sum\limits_{m = 0}^{h}{B_{{{2}^{s}}ml}^{2n+1}}$ and $\sum\limits_{m = 0}^{h}{C_{{{2}^{s}}ml}^{2n+1}}$. We apply a simple relation between the balancing polynomials and the Chebyshev polynomials to further obtain the divisibility properties of $\sum\limits_{m = 0}^{h}{U_{{{2}^{s}}ml-1}^{2n+1}\left(x \right)}$ and $\sum\limits_{m = 0}^{h}{T_{2ml}^{2n+1}\left(x \right)}$ in the Corollaries 5 and 6. This paper can help us to investigate the properties of polynomials and explore further relations between polynomials.

Use of AI tools declaration

The author declares he/she has not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

The author would like to express his/her sincere thanks to anonymous reviewers for their helpful comments and suggestions.

This work is supported by the N. S. F. (12126357) of P. R. China.

Conflict of interest

The author declares no conflict of interest.