#### Fourier series of higher-order Bernoulli functions and their applications

Kim et al. Journal of Inequalities and Applications
Fourier series of higher-order Bernoulli functions and their applications
Taekyun Kim 0 1
Dae San Kim 0
Seog-Hoon Rim 0
Dmitry V. Dolgy 0
0 Kwangwoon University , Seoul
1 Department of Mathematics
the Fourier series of them. k= k= n
Fourier series; Bernoulli polynomials; Bernoulli functions
1 Introduction
ext =
(see [–]).
Bn(x) =
Blxn–l ∈ Q[x] (n ≥ ),
(B + )n – Bn =
if n = ,
B = ,
dBn(x)
Bk xn–k =
Bk xn–k– = n
Bk (n – k)xn–k–
Bk xn––k
= n
(n – )!
(n – k – )!k!
Thus, by (.), we get
For any real number x, we define
x = x – [x] ∈ [, ),
(n ≥ ).
where [x] is the integral part of x. Then Bn( x ) are functions defined on (–∞, ∞) and
periodic with period , which are called Bernoulli functions. The Fourier series for Bm( x )
is given by
Bm x = –m!
= –m!
= –m!
N– ∞
k= nn==–∞ (π in)m
nn==–∞ (π in)m k=
= –m!N –m
l=l=–∞
(x ∈/ Z).
ext =
(see [, , , ]).
2 Fourier series of higher-order Bernoulli functions and their applications
From (.), we note that
B(mr)(x + )
n=–∞
e(x+)t =
B(mr–)(x)
(r–)
mBm– (x) + B(mr)(x)
B(mr)(x)
(r–)
B(mr)() = B(mr)() + mBm– () (m ≥ ).
(r)
Now, we assume that m ≥ , r ≥ . Bm ( x ) is piecewise C∞. Further, in view of (.),
(i =
(r) (r)
Bm+() – Bm+() +
= B(mr–)() +
Replacing m by m – in (.), we get
C(r,m–) – m – (r–)
n Bm–
xe–πinx dx + B
(r)
C(r,m–) – m – (r–)
n Bm–
m(m – ) (r–)
Bm– –
m(m – ) (r–)
Bm– –
m(m – )(m – ) · · ·
From (.) and (.), we can derive equation (.):
x dx =
B(mr)(x) dx =
(r) (r)
Bm+() – Bm+() = B(mr–).
Before proceeding, we recall the following equations:
Bm x = –m!
, (m ≥ ) (see []),
for x ∈ Z, (see [, ]).
x =
n=–∞
= B(mr–) –
= B(mr–) +
= B(mr–) +
for x ∈ Z
m m (r–)
k= k Bm–k Bk( x ) for x ∈/ Z,
x =
(r–)
Bm–k Bk x , for all x ∈ (–∞, ∞).
Therefore, we obtain the following theorem.
(r–)
Theorem . Let m ≥ , r ≥ . Assume that Bm– () = .
(r)
(a) Bm ( x ) has the Fourier series expansion
= B(mr–)() –
m m (r–)
k= k Bm–k Bk( x ), for all x ∈ (–∞, ∞), where Bk( x ) is the Bernoulli
k=
Bm () + m B(mr––)(), for x ∈ Z. Thus we obtain the following theorem.
(r)
(a) B(mr–)() –
nn==–∞ k= (π in)k
Here the convergence is pointwise,
where Bk( x ) is the Bernoulli function.
Bm = –(m)!
= – ((πm))m! n= nm
∞ (–)m = (–)m+ (m)!
(π )m ζ (m).
3 Results and discussion
4 Conclusion
Bm + m B(mr––) for x ∈ Z. In addition, the Fourier series of the higher-order Bernoulli
func(r)
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to the manuscript and typed, read, and approved the final manuscript.
Acknowledgements
This paper is supported by grant NO 14-11-00022 of Russian Scientific Fund.
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