Fourier series of higher-order Bernoulli functions and their applications

Journal of Inequalities and Applications, Jan 2017

In this paper, we study the Fourier series related to higher-order Bernoulli functions and give new identities for higher-order Bernoulli functions which are derived from the Fourier series of them. MSC: 11B68, 42A16.

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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. Bm = –(m)! = – ((πm))m!  n= nm ∞ (–)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. 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Taekyun Kim, Dae Kim, Seog-Hoon Rim, Dmitry Dolgy. Fourier series of higher-order Bernoulli functions and their applications, Journal of Inequalities and Applications, 2017, 8, DOI: 10.1186/s13660-016-1282-y