Abstract
In this paper, we consider sums of finite products of Chebyshev polynomials of the second kind and of Fibonacci polynomials and derive Fourier series expansions of functions associated with them. From these Fourier series expansions, we can express those sums of finite products in terms of Bernoulli polynomials and obtain some identities by using those expressions.
Similar content being viewed by others
1 Introduction and preliminaries
The Chebyshev polynomials \(T_{n}(x)\) of the first kind, the Chebyshev polynomials \(U_{n}(x)\) of the second kind, and the Fibonacci polynomials \(F_{n}(x)\) are respectively defined by the recurrence relations as follows (see [13–15]):
When \(x=1\), \(F_{n}=F_{n}(1)\) (\(n\geq0\)) is the Fibonacci sequence.
From (1.1), (1.2), and (1.3), it can be easily shown that the generating functions for \(T_{n}(x)\), \(U_{n}(x)\), and \(F_{n}(x)\) are respectively given by (see [13–15]):
As is well known, the Bernoulli polynomials \(B_{n}(x)\) are defined by the generating function
For any real number x, we let
denote the fractional part of x, where \([x]\) indicates the greatest integer ≤x.
For any integers m, r with \(m,r\geq1\), we put
where the sum runs over all nonnegative integers \(i_{1},i_{2},\ldots ,i_{r+1}\) with \(i_{1}+i_{2}+\cdots+i_{r+1}=m\).
Then we will consider the function \(\alpha_{m,r}(\langle x\rangle)\) and derive their Fourier series expansions. As a corollary to these Fourier series expansions, we will be able to express \(\alpha_{m,r}(x)\) in terms of Bernoulli polynomials \(B_{n}(x)\). Indeed, our result here is as follows.
Theorem A
For any integers m, r with \(m,r\geq1\), we let
Then we have the identity
Here \((x)_{r}=x(x-1)\cdots(x-r+1)\) for \(r\geq1\), and \((x)_{0}=1\).
Also, for any integers m, r with \(m\geq1\), \(r\geq2\), we let
where the sum runs over all nonnegative integers \(i_{1},i_{2},\ldots,i_{r}\) with \(i_{1}+i_{2}+\cdots+i_{r}=m\).
Then we will consider the function \(\beta_{m,r}(\langle x\rangle)\) and derive their Fourier series expansions. Again, as an immediate corollary to these, we can express \(\beta_{m,r}(x)\) as a linear combination of Bernoulli polynomials. In detail, our result is as follows.
Theorem B
For any integers m, r with \(m\geq1\), \(r\geq2\), we let
Then we have the identity
One particular thing we have to note here is that neither \(U_{n}(x)\) nor \(F_{n}(x)\) is Appell polynomials, while all our related results so far have been only about Appell polynomials (see [1, 5–8]).
Moreover, we will get some interesting identities that follow from Theorems A and B together with Lemmas 1 and 2 in [9].
As was mentioned in [7], studying these kinds of sums of finite products of special polynomials can be well justified by the following. Let us put
Then from the Fourier series expansion of \(\gamma_{m}(\langle x\rangle)\) we can express \(\gamma_{m}(x)\) in terms of Bernoulli polynomials just as in (1.10) and (1.12). Then, after some simple modification of this expression, we are able to obtain the famous Faber–Pandharipande–Zagier identity (see [3]) and some slightly different variant of Miki’s identity (see [2, 4, 10, 12]). For the details on this, the reader is referred to Introduction of the paper [7]. For some related results, we let the reader refer to the papers [1, 5–8].
2 Fourier series expansions for functions associated with Chebyshev polynomials of the second kind
By differentiating equation (1.5) it was shown in [15] and mentioned in [13] that the sum of products in (1.9) can be neatly expressed as in the following. This will play a crucial role in this paper.
Lemma 2.1
Let n, r be nonnegative integers. Then we have the identity
where the sum runs over all nonnegative integers \(i_{1},i_{2},\ldots ,i_{r+1}\) with \(i_{1}+i_{2}+\cdots+i_{r+1}=n\).
It is well known that the Chebyshev polynomials of the second kind \(U_{n}(x)\) are explicitly given by (see [11, 13])
The rth derivative of (2.1) is given by
Then, combining (2.1) and (2.3), we obtain
As in (1.9), we let
where \(m,r\geq1\), and the sum runs over all nonnegative integers \(i_{1},i_{2},\ldots,i_{r+1}\) with \(i_{1}+i_{2}+\cdots+i_{r+1}=m\).
Then we will consider the function
defined on \({\mathbb{R}}\), which is periodic with period 1.
The Fourier series of \(\alpha_{m,r}(\langle x\rangle)\) is
where
For \(m,r\geq1\), we put
Then, for (2.4) and (2.8), we get
where we note that
Now, using (2.1), we note the following:
Thus we have shown that
Replacing m by \(m+1\) and r by \(r-1\), from (2.11) we have
and
We are now ready to determine the Fourier coefficients \(A_{n} ^{(m)}\).
Case 1: \(n\neq0\).
Case 2: \(n=0\).
Before proceeding further, we recall here that
-
(a)
for \(m\geq2\),
$$ B_{m}\bigl(\langle x\rangle\bigr)=-m!\sum _{\substack{n=-\infty\\n\neq0}} ^{\infty}\frac {e^{2\pi inx}}{(2\pi in)^{m}}, $$(2.17) -
(b)
for \(m=1\),
$$ -\sum_{\substack{n=-\infty\\n\neq0}} ^{\infty} \frac{e^{2\pi inx}}{2\pi in}= \textstyle\begin{cases} B_{1}(\langle x\rangle)&\text{for }x\in{\mathbb{R}}-{\mathbb{Z}}, \\ 0&\text{for }x\in{\mathbb{Z}}. \end{cases} $$(2.18)
From (2.15)–(2.18), we now obtain the Fourier series of \(\alpha_{m,r}(\langle x\rangle)\) given by
\(\alpha_{m,r}(\langle x\rangle)\) (\(m,r\geq1\)) is piecewise \(C^{\infty}\). Moreover, \(\alpha_{m,r}(\langle x\rangle)\) is continuous for those positive integers m, r with \(\Delta_{m,r}=0\), and discontinuous with jump discontinuities at integers for those positive integers m, r with \(\Delta_{m,r}\neq0\). Thus, for \(\Delta_{m,r}=0\), the Fourier series of \(\alpha_{m,r}(\langle x\rangle)\) converges uniformly to \(\alpha_{m,r}(\langle x\rangle)\). On the other hand, for \(\Delta_{m,r}\neq0\), the Fourier series of \(\alpha_{m,r}(\langle x\rangle)\) converges pointwise to \(\alpha_{m,r}(\langle x\rangle)\) for \(x\in{\mathbb{R}}-{\mathbb{Z}}\), and converges to
for \(x\in{\mathbb{Z}}\).
From these observations together with (2.19) and (2.20), we have the next two theorems.
Theorem 2.2
For any integers m, r with \(m,r\geq1\), we let
Assume that \(\Delta_{m,r}=0\) for some positive integers \(m,r\). Then we have the following:
-
(a)
$$ \sum_{i_{1}+i_{2}+\cdots+i_{r+1}=m}U_{i_{1}}\bigl(\langle x\rangle \bigr)U_{i_{2}}\bigl(\langle x\rangle\bigr)\cdots U_{i_{r+1}}\bigl( \langle x\rangle\bigr) $$
has the Fourier series expansion
$$\begin{aligned} &\sum_{i_{1}+i_{2}+\cdots+i_{r+1}=m}U_{i_{1}}\bigl(\langle x\rangle \bigr)U_{i_{2}}\bigl(\langle x\rangle\bigr)\cdots U_{i_{r+1}}\bigl( \langle x\rangle\bigr) \\ &\quad =\frac{1}{2r}\Delta_{m+1,r-1}-\sum _{\substack{n=-\infty\\n\neq 0}} ^{\infty} \Biggl(\frac{1}{2r}\sum _{j=1} ^{m} \frac {2^{j}(r+j-1)_{j}}{(2\pi in)^{j}}\Delta_{m-j+1,r+j-1} \Biggr)e^{2\pi inx} \end{aligned}$$for all \(x\in{\mathbb{R}}\), where the convergence is uniform.
-
(b)
$$\begin{aligned} &\sum_{i_{1}+i_{2}+\cdots+i_{r+1}=m}U_{i_{1}}\bigl(\langle x\rangle \bigr)U_{i_{2}}\bigl(\langle x\rangle\bigr)\cdots U_{i_{r+1}}\bigl( \langle x\rangle\bigr) \\ &\quad =\frac{1}{2r}\sum_{\substack{j=0\\j\neq1}} ^{m} 2^{j}\binom {r+j-1}{r-1}\Delta_{m-j+1,r+j-1}B_{j}\bigl( \langle x\rangle\bigr) \end{aligned}$$
for all \(x\in{\mathbb{R}}\).
Theorem 2.3
For any integers m, r with \(m,r\geq1\), we let
Assume that \(\Delta_{m,r}\neq0\) for some positive integers m, r. Then we have the following:
-
(a)
$$\begin{aligned} &\frac{1}{2r}\Delta_{m+1,r-1}-\sum_{\substack{n=-\infty\\n \neq 0}} ^{\infty} \Biggl(\frac{1}{2r}\sum_{j=1} ^{m} \frac {2^{j}(r+j-1)_{j}}{(2\pi in)^{j}}\Delta_{m-j+1,r+j-1} \Biggr)e^{2\pi inx} \\ &\quad = \textstyle\begin{cases} \sum_{i_{1}+i_{2}+\cdots+i_{r+1}=m}U_{i_{1}}(\langle x\rangle)U_{i_{2}}(\langle x\rangle)\cdots U_{i_{r+1}}(\langle x\rangle)&\textit{for }x\in\mathbb{R}-\mathbb{Z}, \\ \frac{1}{2}\Delta_{m,r}& \textit{for }x\in{\mathbb{Z}} \textit{ and }m \textit{ odd}, \\ (-1)^{\frac{m}{2}}\binom{\frac{m}{2}+r}{\frac{m}{2}}+\frac {1}{2}\Delta_{m,r} &\textit{for }x\in{\mathbb{Z}}\textit{ and }m\textit{ even}. \end{cases}\displaystyle \end{aligned}$$
-
(b)
$$\begin{aligned} &\frac{1}{2r}\sum_{j=0} ^{m} 2^{j} \binom{r+j-1}{r-1}\Delta _{m-j+1,r+j-1}B_{j}\bigl( \langle x\rangle\bigr) \\ &\quad =\sum_{i_{1}+i_{2}+\cdots+i_{r+1}=m}U_{i_{1}}\bigl(\langle x \rangle\bigr)U_{i_{2}}\bigl(\langle x\rangle\bigr)\cdots U_{i_{r+1}} \bigl(\langle x\rangle\bigr)\quad \textit{for }x\in{\mathbb{R}}-{\mathbb{Z}}; \\ &\frac{1}{2r}\sum_{\substack{j=0\\j\neq1}} ^{m} 2^{j} \binom {r+j-1}{r-1}\Delta_{m-j+1,r+j-1}B_{j}\bigl( \langle x\rangle\bigr) \\ &\quad = \textstyle\begin{cases} \frac{1}{2}\Delta_{m,r}&\textit{for }x\in\mathbb{Z}\textit{ and }m \textit{ odd}, \\ (-1)^{\frac{m}{2}}\binom{\frac{m}{2}+r}{\frac{m}{2}}+\frac {1}{2}\Delta_{m,r}&\textit{for }x\in{\mathbb{Z}}\textit{ and }m\textit{ even}. \end{cases}\displaystyle \end{aligned}$$
From Theorems 2.2 and 2.3, we immediately obtain the stated result in Theorem A expressing \(\alpha_{m,r}(x)\) as a linear combination of Bernoulli polynomials.
3 Fourier series expansions for functions associated with Fibonacci polynomials
The following lemma is stated as equation (7) in [14] which is important for our purpose.
Lemma 3.1
Let n, r be integers with \(n\geq0\), \(r\geq1\). Then we have the identity
where the sum runs over all nonnegative integers \(i_{1},i_{2},\ldots,i_{r}\) with \(i_{1}+i_{2}+\cdots+i_{r}=n\).
An explicit expression for \(F_{n+1}(x)\) (\(n\geq0\)) is stated in equation (9) of [14].
As was noted in (10) of [14], the \((r-1)\)th derivative of \(F_{n+r}(x)\) is
In addition, it was also noted in [14] that, combining (3.1) and (3.3), we have
As in (1.11), we let
where \(m\geq1\), \(r\geq2\), and the sum runs over all nonnegative integers \(i_{1},i_{2},\ldots,i_{r}\) with \(i_{1}+i_{2}+\cdots+i_{r}=m\).
Then we will consider the function
defined on \({\mathbb{R}}\), which is periodic with period 1.
The Fourier series of \(\beta_{m,r}(\langle x\rangle)\) is
where
For \(m\geq1\), \(r\geq2\), we set
Then, from (3.4) and (3.8), we have
In particular, we note that \(\Omega_{m,r}>0\) for any \(m\geq1\), \(r\geq 2\). Also, we note that
Now, using (3.1), we see the following:
Thus we have shown that
Replacing m by \(m+1\) and r by \(r-1\), from (3.11) we get
and
We are now going to determine the Fourier coefficients \(B_{n} ^{(m)}\).
Case 1: \(n\neq0\).
Case 2: \(n=0\).
From (2.17), (2.18), (3.15), and (3.16), we now have the following Fourier series expansion of \(\beta_{m,r}(\langle x\rangle)\) given by
\(\beta_{m,r}(\langle x\rangle)\) (\(m\geq1\), \(r\geq2\)) is piecewise \(C^{\infty}\) and discontinuous with jump discontinuities at integers, as \(\Omega _{m,r}>0\) for any \(m\geq1\), \(r\geq2\). Thus the Fourier series of \(\beta_{m,r}(\langle x\rangle)\) converges pointwise to \(\beta_{m,r}(\langle x\rangle)\) for \(x\in{\mathbb{R}}-{\mathbb{Z}}\), and converges to
for \(x\in{\mathbb{Z}}\).
From these observations together with (3.17) and (3.18), we have the following theorem.
Theorem 3.2
For any integers m, r with \(m\geq1\), \(r\geq2\), we let
Then we have the following:
-
(a)
$$\begin{aligned} &\frac{1}{r-1}\Omega_{m+1,r-1}-\sum_{\substack{n=-\infty\\n\neq 0}} ^{\infty} \Biggl(\frac{1}{r-1}\sum_{j=1} ^{m} \frac {(r-2+j)_{j}}{(2\pi in)^{j}}\Omega_{m-j+1,r+j-1} \Biggr)e^{2\pi inx} \\ &\quad = \textstyle\begin{cases} \sum_{i_{1}+i_{2}+\cdots+i_{r}=m}F_{i_{1}+1}(\langle x\rangle)F_{i_{2}+1}(\langle x\rangle)\cdots F_{i_{r}+1}(\langle x\rangle)&\textit{for }x\in{\mathbb{R}}-{\mathbb{Z}}, \\ \frac{1}{2}\Omega_{m,r}& \textit{for }x\in{\mathbb{Z}}\textit{ and }m\textit{ odd}, \\ \binom{\frac{m}{2}+r-1}{\frac{m}{2}}+\frac{1}{2}\Omega_{m,r}& \textit{for }x\in{\mathbb{Z}}\textit{ and }m\textit{ even}. \end{cases}\displaystyle \end{aligned}$$
-
(b)
$$\begin{aligned} &\frac{1}{r-1}\sum_{j=0} ^{m} \binom{r-2+j}{j}\Omega _{m-j+1,r+j-1}B_{j}\bigl(\langle x\rangle \bigr) \\ &\quad =\sum_{i_{1}+i_{2}+\cdots+i_{r}=m}F_{i_{1}+1}\bigl(\langle x \rangle\bigr)F_{i_{2}+1}\bigl(\langle x\rangle\bigr)\cdots F_{i_{r}+1} \bigl(\langle x\rangle\bigr)\quad \textit{for }x\in{\mathbb{R}}-{\mathbb{Z}}; \\ &\frac{1}{r-1}\sum_{\substack{j=0\\j\neq1}} ^{m} \binom {r-2+j}{j}\Omega_{m-j+1,r+j-1}B_{j}\bigl(\langle x\rangle\bigr) \\ &\quad = \textstyle\begin{cases} \frac{1}{2}\Omega_{m,r}&\textit{for }x\in{\mathbb{Z}}\textit{ and }m \textit{ odd}, \\ \binom{\frac{m}{2}+r-1}{\frac{m}{2}}+\frac{1}{2}\Omega _{m,r}&\textit{for }x\in{\mathbb{Z}}\textit{ and }m\textit{ even}. \end{cases}\displaystyle \end{aligned}$$
From Theorem 3.2, we immediately get the result in Theorem B expressing \(\beta_{m,r}(x)\) as a linear combination of Bernoulli polynomials.
4 Applications
Let \(T_{n}(x)\) (\(n\geq0\)) be the Chebyshev polynomials of the first kind given by (1.1) or (1.4). We need the following lemma from [9].
Lemma 4.1
([9, Lemmas 1, 2])
Let \(n\geq0\), \(m\geq1\) be integers. Then we have the following:
Substituting \(x=\frac{\sqrt{-1}}{2}\) into (1.10) and using (4.1), we have
On the other hand, with \(x=1\) and replacing r by \(r+1\), from (1.12) we obtain
Combining (4.4) and (4.5), we get
Substituting \(T_{a} (\frac{\sqrt{-1}}{2} )\) for x in (1.10) and using (4.2), we get: for any positive integer a,
Finally, replacing x by \(T_{a}(x)\) in (1.10) and using (4.3), we have: for any positive integer a,
5 Results and discussion
In this paper, we study sums of finite products of Chebyshev polynomials of the second kind and of Fibonacci polynomials and derive Fourier series expansions of functions associated with them. From these Fourier series expansions, we can express those sums of finite products in terms of Bernoulli polynomials and obtain some identities by using those expressions. The Fourier series expansion of the Chebyshev polynomials and Fibonacci polynomials are useful in computing the special values of zeta function or some special functions (see [5, 7, 9, 11, 13–15]). It is expected that the Fourier series of those polynomials will find some applications in relationship to the generalizations of the special zeta functions.
6 Conclusion
In this paper, we considered the Fourier series expansions of functions associated with Chebyshev polynomials of the second kind and of Fibonacci polynomials. The Fourier series are determined completely.
References
Agarwal, R.P., Kim, D.S., Kim, T., Kwon, J.: Sums of finite products of Bernoulli functions. Adv. Differ. Equ. 2017, 237 (2017)
Dunne, G.V., Schubert, C.: Bernoulli number identities from quantum field theory and topological string theory. Commun. Number Theory Phys. 7, 225–249 (2013)
Faber, C., Pandharipande, R.: Hodge integrals and Gromov–Witten theory. Invent. Math. 139, 173–199 (2000)
Gessel, I.M.: On Miki’s identity for Bernoulli numbers. J. Number Theory 110, 75–82 (2005)
Kim, T., Kim, D.S., Jang, G.-W., Kwon, J.: Fourier series of finite products of Bernoulli and Genocchi functions. J. Inequal. Appl. 2017, 157 (2017)
Kim, T., Kim, D.S., Jang, G.W., Kwon, J.: Sums of finite products of Euler functions. In: Advances in Real and Complex Analysis with Applications. Trends in Mathematics, pp. 243–260. Springer, Berlin (2017)
Kim, T., Kim, D.S., Jang, L.-C., Jang, G.-W.: Fourier series of sums of products of Bernoulli functions and their applications. J. Nonlinear Sci. Appl. 10(5), 2798–2815 (2017)
Kim, T., Kim, D.S., Jang, L.C., Jang, G.-W.: Sums of finite products of Genocchi functions. Adv. Differ. Equ. 2017, 268 (2017)
Ma, R., Zhang, W.: Several identities involving the Fibonacci numbers and Lucas numbers. Fibonacci Q. 45(2), 164–170 (2007)
Miki, H.: A relation between Bernoulli numbers. J. Number Theory 10, 297–302 (1978)
Prodinger, H.: Representing derivatives of Chebyshev polynomials by Chebyshev polynomials and related questions. Open Math. 15, 1156–1160 (2017)
Shiratani, K., Yokoyama, S.: An application of p-adic convolutions. Mem. Fac. Sci., Kyushu Univ., Ser. A, Math. 36, 73–83 (1982)
Wang, S.: Some new identities of Chebyshev polynomials and their applications. Adv. Differ. Equ. 2015, 355 (2015)
Yuan, Y., Zhang, W.: Some identities involving the Fibonacci polynomials. Fibonacci Q. 40, 314–318 (2002)
Zhang, W.: Some identities involving the Fibonacci numbers and Lucas numbers. Fibonacci Q. 42, 149–154 (2004)
Funding
This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (No. NRF-2017R1D1A1B03034892).
Author information
Authors and Affiliations
Contributions
All authors contributed equally to the manuscript and typed, read, and approved the final manuscript.
Corresponding author
Ethics declarations
Competing interests
The authors declare that they have no competing interests.
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Kim, T., Kim, D.S., Dolgy, D.V. et al. Sums of finite products of Chebyshev polynomials of the second kind and of Fibonacci polynomials. J Inequal Appl 2018, 148 (2018). https://doi.org/10.1186/s13660-018-1744-5
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13660-018-1744-5