Fourier series of higher-order Daehee and Changhee functions and their applications

In the paper, the author considers the Fourier series related to higher-order Daehee and Changhee functions and establishes some new identities for higher-order Daehee and Changhee functions.


Introduction and main results
It is common knowledge that the Bernoulli polynomials B n (x) and the Euler polynomials E n (x) for n ≥  can be generated by By replacing t by e t - in (), we can derive where S  (n, m) is the Stirling number of the second kind which is given by Comparing the coefficients on the both sides of (), we obtain Also, with the viewpoint of deformed Euler polynomials, the Changhee polynomials Ch n (x) for n ≥  are defined by the generating function to be Definition () can be written as Combination of this identity with () results in the following relation: Equating coefficients on the very ends of the above identity leads to E n (x) = n m= Ch m (x)S  (n, m) (n ≥ ).
In recent decades, many mathematicians have investigated some interesting extensions or modifications of the Daehee and Changhee polynomials along with related combinatorial identities and their applications (see [, , , , , , , ]). Especially, Kim and his coauthors have studied the Fourier series related to various types of Bernoulli functions in [, -, ]. The purpose of this paper is to study the Fourier series related to higherorder Daehee and Changhee functions and establish some new identities for higher-order Daehee and Changhee functions.
For any real number x, we define where [x] is the integer part of x. Then D n ( x ) are functions defined on (-∞, ∞) and periodic with period , which are called Daehee functions. For r ∈ N and n ≥ , we note that the higher-order Daehee polynomials D (r) n (x) and the higher-order Changhee polynomials Ch (r) n (x) may also be represented by the following generating function: respectively (see [, , ]). When x = , D (r) n = D (r) n () are called the higher-order Daehee numbers and Ch (r) n = Ch (r) n () are called the higher-order Changhee numbers. And it is easy to see that Then D (r) n ( x ) and Ch (r) n ( x ) are functions defined on (-∞, ∞) and periodic of period , which are called Daehee functions of order r and Changhee functions of order r, respectively.
Recall from [, ] that the Bernoulli function may be represented by The Fourier series expansion of the Bernoulli functions is useful in computing the special values of the Dirichlet L-functions. For details, one is referred to [].
Our main results in this paper can be stated as the following theorems.
Here the convergence is uniform.
Here the convergence is pointwise.
m ( x ) has the Fourier series expansion Here the convergence is uniform.
Here the convergence is pointwise.
where B k ( x ) is the Bernoulli function.

Proofs of Theorems 1-4
We are now in a position to prove our four theorems. By analyzing definition (), we have Furthermore, we observe that Letting x =  in the above equation leads to Further, in view of (), D (r) m ( x ) is continuous for those (r, m) with D (r) m- = , and is discontinuous with jump discontinuities at integers for those (r, m) with D (r) m- = . The Fourier series of D (r) m ( x ) may be represented by Replacing m by m - in (), we arrive at the following result: Case  Let n = . Then we acquire that Moreover, we observe that Combining () with (), we immediately derive the following equation: Case  Let n = . Then we have While that in () converges pointwise, the series in () converges uniformly. We assume that D (r) m- = . Then we have D (r) m () = D (r) m for m ≥ . As D (r) m ( x ) is piecewise C ∞ and continuous, the Fourier series of D (r) m ( x ) converges uniformly to D (r) m ( x ) and () Note that () holds whether D (r) m- =  or not. However, if D (r-) m- = , then Therefore, we obtain the result in Theorem .
Assume next that D (r) m- = . Then we have D (r) m () = D (r) m and hence D (r) m ( x ) is piecewise C ∞ and discontinuous with jump discontinuities at integers. Thus the Fourier series of D (r) m ( x ) converges pointwise to D (r) m ( x ) for x / ∈ Z, and converges to   (D (r) m + D (r) m ()) = D (r) m + (m/)D (r) m- for x ∈ Z. Finally, we obtain the formulas in Theorem . From now on we focus on definition (). Then we can find In other words,

This equation means that
Assume that m ≥  and r ≥  Ch (r) m ( x ) is piecewise C ∞ . In addition, Ch (r) m ( x ) is continuous for those (r, m) with Ch (r) m = Ch (r-) m and discontinuous with jump discontinuities at integers for those (r, m) with Ch (r) Here Case  Let n = . Then we acquire that In addition, we observe that Therefore, we can derive the following equation: Here, we used the fact that Finally, assume that Ch (r) m = Ch (r-) m . Then we have Ch (r) m () = Ch (r) m and hence Ch (r) m ( x ) is piecewise C ∞ and discontinuous with jump discontinuities at integers. Thus the Fourier series of Ch (r) m ( x ) converges pointwise to Ch (r) m ( x ) for x / ∈ Z, and converges to   (Ch (r) m + Ch (r) m ()) = Ch (r-) m for x ∈ Z. From the above considerations, the proof of Theorem  is complete.

Conclusions
In this paper, the author considered the Fourier series expansion of the higher-order Daehee functions D (r) n ( x ) and the higher-order Changhee functions Ch (r) n ( x ) which are obtained by extending by periodicity of period  the higher-order Daehee polynomials D (r) n (x) and the higher-order Changhee polynomials Ch (r) n (x) on [, ), respectively. The Fourier series are explicitly determined. Depending on whether D (r) n ( x ) and Ch (r) n ( x ) are zero or not, the Fourier series of these functions converge uniformly or converge pointwise. In addition, the Fourier series of the higher-order Daehee functions D (r) n ( x ) and the higherorder Changhee functions Ch (r) n ( x ) are expressed in terms of the Bernoulli functions B k ( x ). Thus we established the relations between these functions and Bernoulli functions.