Sums of finite products of Chebyshev polynomials of the second kind and of Fibonacci polynomials

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.


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][14][15]): T n+2 (x) = 2xT n+1 (x) -T n (x) (n ≥ 0), T 0 (x) = 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][14][15]):  (1.9) where the sum runs over all nonnegative integers i 1 , i 2 , . . . , i r+1 with i 1 + i 2 + · · · + i r+1 = m. Then we will consider the function α m,r ( x ) and derive their Fourier series expansions. As a corollary to these Fourier series expansions, we will be able to express α 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 ≥ 1, we let Then we have the identity Here (x) r = x(x -1) · · · (xr + 1) for r ≥ 1, and (x) 0 = 1. Also, for any integers m, r with m ≥ 1, r ≥ 2, we let where the sum runs over all nonnegative integers i 1 , i 2 , . . . , i r with i 1 + i 2 + · · · + i r = m.
Then we will consider the function β m,r ( x ) and derive their Fourier series expansions. Again, as an immediate corollary to these, we can express β 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 ≥ 1, r ≥ 2, 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][6][7][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 γ m ( x ) we can express γ 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].

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.
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 ≥ 1, and the sum runs over all nonnegative integers i 1 , Then we will consider the function where For m, r ≥ 1, 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 (m) n . Case 1: n = 0.
Before proceeding further, we recall here that (a) for m ≥ 2, (2.18) From (2.15)-(2.18), we now obtain the Fourier series of α m,r ( x ) given by is continuous for those positive integers m, r with m,r = 0, and discontinuous with jump discontinuities at integers for those positive integers m, r with m,r = 0. Thus, for m,r = 0, the Fourier series of α m,r ( x ) converges uniformly to α m,r ( x ). On the other hand, for m,r = 0, the Fourier series of α m,r ( x ) converges pointwise to α m,r ( x ) for x ∈ R -Z, and converges to Assume that m,r = 0 for some positive integers m, r. Then we have the following: has the Fourier series expansion for all x ∈ R, where the convergence is uniform.
for all x ∈ R.

Theorem 2.3 For any integers m, r with m, r ≥ 1, we let
Assume that m,r = 0 for some positive integers m, r. Then we have the following:

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.
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 ≥ 1, r ≥ 2, and the sum runs over all nonnegative integers i 1 , i 2 , . . . , i r with i 1 + i 2 + · · · + i r = m. Then we will consider the function In particular, we note that m,r > 0 for any m ≥ 1, r ≥ 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 d dx and β m,r (0) = β m,r (1) ⇐⇒ m,r = 0. (3.14) We are now going to determine the Fourier coefficients B (m) n . Case 1: n = 0. (3.17) is piecewise C ∞ and discontinuous with jump discontinuities at integers, as m,r > 0 for any m ≥ 1, r ≥ 2. Thus the Fourier series of β m,r ( x ) converges pointwise to β m,r ( x ) for x ∈ R -Z, and converges to From these observations together with (3.17) and (3.18), we have the following theorem. Then we have the following:

Applications
Let T n (x) (n ≥ 0) be the Chebyshev polynomials of the first kind given by (1.1) or (1.4). We need the following lemma from [9].

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][14][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.

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.