Fourier series of sums of products of Bernoulli functions and their applications

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1 Available online at J. Nonlinear Sci. Appl., 7, Research Article Journal Homepage: Fourier series of sums of products of Bernoulli functions and their applications Taeyun Kim a,b, Dae San Kim c, Lee-Chae Jang d,, Gwan-Woo Jang b a Department of Mathematics, College of Science, Tianjin Polytechnic University, Tianjin City, 3387, China. b Department of Mathematics, Kwangwoon University, Seoul 39-7, Republic of Korea. c Department of Mathematics, Sogang University, Seoul -74, Republic of Korea. d Graduate School of Education, Konu University, Seoul 43-7, Republic of Korea. Communicated by Y. J. Cho Abstract We consider three types of sums of products of Bernoulli functions and derive their Fourier series expansions. In addition, we express each of them in terms of Bernoulli functions. c 7 All rights reserved. Keywords: Fourier series, Bernoulli polynomials, Bernoulli functions. MSC: B68, S8, 5A9, 5A3.. Introduction Let B m x be the Bernoulli polynomials given by the generating function For any real number x, we let t e t ext m B m x tm, see [,, 7, 9, 5, 7]. m! < x > x [x] [,, denote the fractional part of x. Here we will consider the following three types of sums of products of Bernoulli functions and derive their Fourier series expansions. Further, we will express each of them in terms of Bernoulli functions B m < x >. α m < x > m B < x >B m < x >, m ; β m < x > m!m! B < x >B m < x >, m ; Corresponding author addresses: tim@w.ac.r Taeyun Kim, dsim@sogang.ac.r Dae San Kim, lcjang@onu.ac.r Lee-Chae Jang, jgw5687@naver.com Gwan-Woo Jang doi:.436/jnsa Received 6--9

2 T. K. Kim, D. S. Kim, L.-C. Jang, G.-W. Jang, J. Nonlinear Sci. Appl., 7, γ m < x > m m B < x >B m < x >, m. For elementary facts about Fourier analysis, the reader may refer to any boo for example, see [6, 8, 8]. As to γ m < x >, we note that the following polynomial identity follows immediately from 4.3 and 4.4. m m B x B m x m B m + + m m m m B m B x + m H mb m x, m,. where H m m j j are the harmonic numbers. Simple modification of. yields m m B x B m x + m + m B x B m x m B B m x + m H mb m x. m B x B m, m. Letting x in. gives a slightly different version of the well-nown Mii s identity see [4]: m m B B m m m B B m + m H mb m, m..3 Setting x in.3 with B m m m B m m B m B m, we have m m B B m m m B B m + m H mb m, m,.4 which is the Faber-Pandharipande-Zagier identity see [4]. Some of the different proofs of Mii s identity can be found in [3, 5, 4, 6]. Dunne-Schubert in [3] uses the asymptotic expansion of some special polynomials coming from the quantum field theory computations, Gessel in [5] is based on two different expressions for Stirling numbers of the second ind S n,, Mii in [4] exploits a formula for the Fermat quotient ap a p modulo p, and Shiratani-Yooyama in [6] employs p-adic analysis. As we can see, all of these proofs are quite involved. On the other hand, our proof of Mii s and Faber-Pandharipande-Zagier identities follow from the polynomial identity., which in turn follows immediately the Fourier series expansion results for γ m < x > in Theorems 4.3 and 4.4, together with the elementary manipulations outlined in.-.4. The obvious polynomial identities can be derived also for α m < x > and β m < x > from.6 and.7, and 3.3 and 3.4, respectively. Some related wors can be found in [ 3].

3 T. K. Kim, D. S. Kim, L.-C. Jang, G.-W. Jang, J. Nonlinear Sci. Appl., 7, Fourier series of functions of the first type In this section, we consider the function α m < x > B < x >B m < x >,. defined on, which is periodic of period. The Fourier series of α m < x > is where A m n n A m n e πinx,. α m < x >e πinx dx α m xe πinx dx. Then, we would lie to determine the Fourier coefficients A m n in. Case : n, We note that α mx m + α m x. Indeed, α mx A m n [ αm xe πinx] πin + α πin mxe πinx dx. B xb m x + m B xb m x m B xb m x + m B xb m x m m + B xb m x + m B xb m x m m + B xb m x m + α m x. Hence Observe here that A m n πin α m α m + m + πin Am n. α m α m B m + δ m,..3 Indeed, α m α m B B m B B m

4 T. K. Kim, D. S. Kim, L.-C. Jang, G.-W. Jang, J. Nonlinear Sci. Appl., 7, Thus, we have B + δ, B m + δ,m B B m B δ,m + δ, B m + δ, δ,m B m + δ m,. A m n m + πin Am n πin B m + δ m, m + m πin πin Am n πin B m + δ m,3 πin B m + δ m, m + m πin Am n m + πin B m + δ m,3 πin B m + δ m, { m + m m πin πin Am3 n } πin B m3 + δ m,4 m + πin B m + δ m,3 πin B m + δ m, m + 3 πin 3 Am3 n 3 m + πin B m + δ m,+ and. m + m m m + A πin m n πin B m + δ m,+ m + m m A πin m n A n α xe πinx dx B xe πinx dx x e πinx dx xe πinx dx { πin m + πin B m m + m πin m, e πinx dx [ xe πinx ] + πin e πinx dx } πin. Hence A m n m + m πin m, m m m + πin B m. m + πin B m m + m πin m

5 T. K. Kim, D. S. Kim, L.-C. Jang, G.-W. Jang, J. Nonlinear Sci. Appl., 7, Case : n, Then, we have the following theorem. Theorem.. For n N {}, we have Proof. Let A m { A m m+ B m, for m,, for m, m + I m,n B m + δ m, α m xdx..4 B m xb n xdx,, for m. for m, n. Then I m,n I n,m, I,, I m,, for m. We can show, by integration by parts, that Thus, from. and.4, we have where I m,n m B m+n, for m, n. A m δ m m+n m B xb m xdx, m I,m + I,m + I m, m m B m Bm m m m + B mδ m m + B m + δ m,, {, if m mod,, if m mod. From.3, we observe that α m α m B m + δ m,.

6 T. K. Kim, D. S. Kim, L.-C. Jang, G.-W. Jang, J. Nonlinear Sci. Appl., 7, As B, B n+, for n, and n+ B n >, for n, we see that B m + δ m, B m + δ m, m is an even positive integer m is an odd positive integer. α m < x > is piecewise C. In addition, α m < x > is continuous for all even positive integers m and discontinuous with jump discontinuities at integers for all odd positive integers m. We now recall the following facts about Bernoulli functions B m < x >: a for m, b for m, where Z c is R Z. B m < x > m! n,n n,n e πinx πin m ; e πinx { πin B < x >, for x Z c,, for x Z, Assume first that m is an even positive integer. Then α m α m, and thus α m < x > is piecewise C, and continuous. Hence the Fourier series of α m < x > converges uniformly to α m < x >, and α m < x > m + B m + m + πin B m e πinx m + B m + n,n, m If m 4 and m is an even integer, then we have For m, α m < x > m + B m +, m m +!, m B m! n,n m + B m B < x >! { B < x >, for x Z + B m c,, for x Z, m + B m + m + B m B < x >.! α < x > + B + Thus we obtain the following theorem., m + B B < x >.! Theorem.. Let m be an even positive integer. Then we have the following. a m B < x >B m < x > has the Fourier series expansion B < x >B m < x > m + B m + n,n for all x,, where the convergence is uniform., m m + πin e πinx πin. B m e πinx,.5

7 T. K. Kim, D. S. Kim, L.-C. Jang, G.-W. Jang, J. Nonlinear Sci. Appl., 7, b B < x >B m < x > m + B m +, m m + B m < x > + + m for all x,. Here B < x > is the Bernoulli function. m + B m B < x >! m m + B m B < x >, Assume next that m is an odd positive integer. Then α m α m, and hence α m < x > is piecewise C and discontinuous with jump discontinuities at integers. Thus the Fourier series of α m < x > converges pointwise to α m < x >, for x Z c, and converges to for x Z. Hence we get the following theorem. α m + α m α m + B m + δ m, α m + B m, Theorem.3. Let m be an odd positive integer. Then we have the following:.6 a B m + m + δ m, + n,n, m m + πin B m e πinx { m B < x >B m < x >, for x Z c, m B B m + B m, for x Z. Here the convergence is pointwise. b B m + m + δ m, +, m m + B m B < x >! B < x >B m < x >,.7 for x Z c. B m + m + δ m, +, m B B m + B m, m + B m B < x >! for x Z.

8 T. K. Kim, D. S. Kim, L.-C. Jang, G.-W. Jang, J. Nonlinear Sci. Appl., 7, Fourier series of functions of the second type In this section, we consider the function β m < x >!m! B < x >B m < x >, defined on, which is periodic of period. The Fourier series of β m < x > is where B m n n B m n e πinx, 3. β m < x >e πinx dx β m xe πinx dx. Then, we would lie to determine the Fourier coefficients B m n β mx Case : n, { m!m! B xb m x + in 3.. For this, we observe that m }!m! B xb m x m!m! B xb m x +!m! B xb m x m!m! B xb m x +!m! B xb m x β m x. Observe that β m β m B m n [ βm xe πinx] πin + πin πin β m β m + πin πin β m β m + πin Bm n.!m! B B m B B m β mxe πinx dx β m xe πinx dx!m! {B + δ, B m + δ,m B B m }!m! {B δ,m + B m δ, + δ, δ,m } m! B m + m! B m + m! δ,m m! B m + δ m,. 3.

9 T. K. Kim, D. S. Kim, L.-C. Jang, G.-W. Jang, J. Nonlinear Sci. Appl., 7, Hence, and Thus B m n πin Bm n πin πinm! B m + δ m, πin Bm n πinm! B m + δ m,3 πinm! B m + δ m, m! πin Bm n m! πin m!. πin 3 Bm3 n m! πin πin B m + δ m, πin Bm3 n B m n B n πin B m + δ m,3 πinm 3! B m3 + δ m,4 πin B m + δ m,3 m! m 3! πin 3 B m3 + δ m,4 πin B m + δ m,3 m m! β xe πinx dx B xe πinx dx x e πinx dx πin πin. B m n m πin m, m xe πinx dx m! πin B m + δ m, πin B m + δ m, πin B m + δ m,+, [ xe πinx ] + πin m! e πinx dx e πinx dx πin B m + δ m,+ m πin m m! πin B m πin m m! πin B m.

10 T. K. Kim, D. S. Kim, L.-C. Jang, G.-W. Jang, J. Nonlinear Sci. Appl., 7, Case : n, Then, we have the following theorem. Theorem 3.. For n N {}, we have B m β m xdx. B m { Bm m!, for m mod,, for m mod, Proof. Let B m B m m! + δ m,, for m. β m x!m!!m! I,m B xb m xdx m m! I B,m + m!m! m B m m m! { Bm m!, for m mod,, for m mod. B m m! + δ m,, From 3., we see that β m β m B m + δ m,. As we saw in.5, B m + δ m, B m + δ m, m is an even positive integer m is an odd positive integer. β m < x > is piecewise C. In addition, β m < x > is continuous for all even positive integers m and discontinuous with jump discontinuities at integers for all odd positive integers m. Assume first that m is an even positive integer. Then β m β m, and so β m < x > is piecewise C, and continuous. Hence the Fourier series of β m < x > converges uniformly to β m < x >, and β m < x > B m m! + n,n B m m! + m, m m! πin B m e πinx, m m!! B e m! πinx πin. n,n

11 T. K. Kim, D. S. Kim, L.-C. Jang, G.-W. Jang, J. Nonlinear Sci. Appl., 7, If m 4 and m is an even integer, then we have For m, β m < x > B m m! + m! Now, we obtain the following theorem. + m! m B m m! + m! β < x > B! +!, m B m,, m m B m B < x > { B < x >, for x Z c,, for x Z m B m B < x >. B B < x >. Theorem 3.. Let m be an even positive integer. Then we have the following. a m!m! B < x >B m < x > has the Fourier expansion b!m! B < x >B m < x > B m m! + n,n for all x,, where the convergence is uniform.!m! B < x >B m < x > B m m! + m!, m for all x,. Here B < x > is the Bernoulli function., m m! m πin B m e πinx, B m B < x >, 3.3 Assume next that m is an odd positive integer. Then β m β m, and hence β m < x > is piecewise C and discontinuous with jump discontinuities at integers. Thus the Fourier series of β m < x > converges pointwise to β m < x >, for x Z c and converges to β m + β m β m + β m + m! B m + δ m, β m + m! B m, for x Z. Hence we have the following theorem. Theorem 3.3. Let m be an odd positive integer. Then we have the following. a B m m! + δ m, + Here the convergence is pointwise. n,n, m m! πin B m e πinx { m!m! B < x >B m < x >, for x Z c, m!m! B B m + m! B m, for x Z.

12 T. K. Kim, D. S. Kim, L.-C. Jang, G.-W. Jang, J. Nonlinear Sci. Appl., 7, b B m m! + δ m, + m!, m m B m B < x >!m! B < x >B m < x >, 3.4 for x Z c. B m m! + δ m, + m!, m m B m B < x >!m! B B m + m! B m, for x Z. 4. Fourier series of functions of the third type In this section, we consider the function γ m < x > m m B < x >B m < x >, defined on, which is periodic of period. The Fourier series of γ m < x > is where C m n n γ m < x >e πinx dx Next, we would lie to determine the Fourier coefficients C m n γ mx m m m C m n e πinx, 4. γ m xe πinx dx. in 4.. For this, we observe that m {B xb m x + m B xb m x} m m B xb m x + B xb m x m m B xb m x + m B mx + m m B mx + m γ m x. B xb m x m + B xb m x

13 T. K. Kim, D. S. Kim, L.-C. Jang, G.-W. Jang, J. Nonlinear Sci. Appl., 7, Case : n, Observe that C m n [ γm xe πinx] πin + πin πin γ m γ m + πinm γ mxe πinx dx B m xe πinx dx + m πin Cm n. and γ m γ m m m m B m xe πinx dx m m B B m B B m m B + δ, B m + δ,m B B m m B δ,m + δ, B m + δ, δ,m m B m + m B m + m δ m, m B m + m δ m, m B m + δ m,, [ Bm xe πinx] + πin m m B m B m + πin m B m xe πinx dx B m xe πinx dx. 4. Denoting B mxe πinx dx by ρ m n, we have ρ n m m πin ρm n πin δ m, m m πin πin ρm n πin δ m, πin δ m, mm πin ρm n m πin δ m, πin δ m, mm m πin πin ρm3 m πin δ m,3 m πin δ m, πin δ m, mm m πin 3 ρ m3 mm m πin 3 δ m,3 m πin δ m, πin δ m,. m m! m ρ πin m n πin δ m, m! πin m, as one can see that ρ n πin. Thus, for n, C m n m πin Cm n πinm B m! m + δ m, πin m

14 T. K. Kim, D. S. Kim, L.-C. Jang, G.-W. Jang, J. Nonlinear Sci. Appl., 7, and m πin m πin Cm n πinm B m 3! m + δ m,3 πin m m! πin m πinm B m + δ m, m m πin C m n m m πin B m + δ m,3 m πin B m m 3! m + δ m, πin m m m m 3 πin πin Cm3 n m m! πin m πinm 3 B m3 + δ m,4 m πin B m + δ m,3 m πin B m + δ m, m! πin m m m m 3 πin 3 C m3 m m n m 3πin 3 B m3 + δ m,4 m m πin B m + δ m,3 m πin B m + δ m, m m m 4! m m 3! m! πin m πin m πin m. We note that m m! C πin m n c n γ xe πinx dx B x e πinx dx [ x x + πin 4 πin πin πin. m m 4! πin m m m 3! πin m m m m πin B m! n + δ m,+ πin m x x + e πinx dx 4 ] e πinx + x e πinx dx πin [ x e πinx ] + πin e πinx dx m m H m, m, where H m m is the harmonic number. Thus we have the following theorem. Theorem 4.. For n, C m m m! n πin m H m m m πin B m + δ m,+.

15 T. K. Kim, D. S. Kim, L.-C. Jang, G.-W. Jang, J. Nonlinear Sci. Appl., 7, Case : n, Then, we have the following theorem. Theorem 4.. For m N {}, we have Proof. Let C m C m { m B m, for m mod, for m mod C m m m γ m xdx m γ m xdx. B xb m xdx m m I,m m B m m m { B m m, for m mod,, for m mod, B m m + δ m,. B m m + δ m,, for m. B m m m From 4., we see that As we saw in.5, γ m γ m m B m + δ m,. B m + δ m, B m + δ m, m is an even positive integer m is an odd positive integer. γ m < x > is piecewise C. In addition, γ m < x > is continuous for all even positive integers m and discontinuous with jump discontinuities at integers for all odd positive integers m. Assume first that m is an even positive integer. Then γ m γ m, and thus γ m < x > is piecewise C, and continuous. Hence the Fourier series of γ m < x > converges uniformly to γ m < x >, and γ m < x > m B m + n,n { m B m + m H m m! + m m m m m m! πin m H m B m n,n! e πinx πin m n,n m m πin B m e πinx πin. } e πinx

16 T. K. Kim, D. S. Kim, L.-C. Jang, G.-W. Jang, J. Nonlinear Sci. Appl., 7, If m 4 and m is an even integer, then we have m γ m < x > m B m + m H mb m < x > + m m + { m B < x >, for x Z B m c, m m, for x Z, m B m + m H mb m < x > + m m m m for all x,. For m, Hence, we obtain the following theorem. γ < x > B + H B < x >. Theorem 4.3. Let m be an even positive integer. Then we have the following. a m m B < x >B m < x > has the Fourier expansion b m B < x >B m < x > m B m + n,n for all x,, where the convergence is uniform. m B m B < x > B m B < x >, { } m m! πin m H m m m πin B m e πinx, m B < x >B m < x > m B m + m H mb m < x > + m m m m B m B < x >, 4.3 for all x,. Here B < x > is the Bernoulli function. Assume next that m is an odd positive integer. Then γ m γ m, and hence γ m < x > is piecewise C and discontinuous with jump discontinuities at integers. Thus the Fourier series of γ m < x > converges pointwise to γ m < x >, for x Z c and converges to γ m + γ m γ m + γ m + m B m + m δ m, γ m + m B m, for x Z. Hence we have the following theorem. Theorem 4.4. Let m be an odd positive integer. Then we have the following.

17 T. K. Kim, D. S. Kim, L.-C. Jang, G.-W. Jang, J. Nonlinear Sci. Appl., 7, a m B m + { } m δ m! m, + πin m H m m m πin B m n,n { m m B < x >B m < x >, for x Z c, m m B B m + m B m, for x Z. e πinx b Here the convergence is pointwise. for x Z c. for x Z. m B m + δ m, + m H mb m < x > + m m m m B < x >B m < x >, m B m + δ m, + m H mb m < x > + m m m m B B m + m B m, m m m B m B < x > m B m B < x > 4.4 References [] A. Bayad, T. Kim, Higher recurrences for Apostol-Bernoulli-Euler numbers, Russ. J. Math. Phys., 9,. [] D. Ding, J.-Z. Yang, Some identities related to the Apostol-Euler and Apostol-Bernoulli polynomials, Adv. Stud. Contemp. Math. Kyungshang,, 7. [3] G. V. Dunne, C. Schubert, Bernoulli number identities from quantum field theory and topological string theory, Commun. Number Theory Phys., 7 3, [4] C. Faber, R. Pandharipande, Hodge integrals and Gromov-Witten theory, Invent. Math., 39, [5] I. M. Gessel, On Mii s identity for Bernoulli numbers, J. Number Theory, 5, [6] L. C. Jang, T. Kim, D. J. Kang, A note on the Fourier transform of fermionic p-adic integral on Z p, J. Comput. Anal. Appl., 9, [7] T. Kim, q-bernoulli numbers and polynomials associated with Gaussian binomial coefficients, Russ. J. Math. Phys., 5 8, [8] T. Kim, A note on the Fourier transform of p-adic q-integrals on Z p, J. Comput. Anal. Appl., 9, [9] T. Kim, Some identities for the Bernoulli, the Euler and the Genocchi numbers and polynomials, Adv. Stud. Contemp. Math. Kyungshang,, 3 8. [] D. S. Kim, T. Kim, Bernoulli basis and the product of several Bernoulli polynomials, Int. J. Math. Math. Sci.,, pages. [] D. S. Kim, T. Kim, A note on higher-order Bernoulli polynomials, J. Inequal. Appl., 3 3, 9 pages. [] D. S. Kim, T. Kim, Some identities of higher order Euler polynomials arising from Euler basis, Integral Transforms Spec. Funct., 4 3, [3] D. S. Kim, T. Kim, S.-H. Lee, D. V. Dolgy, S.-H. Rim, Some new identities on the Bernoulli and Euler numbers, Discrete Dyn. Nat. Soc.,, pages. [4] H. Mii, A relation between Bernoulli numbers, J. Number Theory, 978, 97 3., [5] K. Shiratani, On some relations between Bernoulli numbers and class numbers of cyclotomic fields, Mem. Fac. Sci. Kyushu Univ. Ser. A, 8 964, [6] K. Shiratani, S. Yooyama, An application of p-adic convolutions, Mem. Fac. Sci. Kyushu Univ. Ser. A, 36 98,

18 T. K. Kim, D. S. Kim, L.-C. Jang, G.-W. Jang, J. Nonlinear Sci. Appl., 7, [7] L. C. Washington, Introduction to cyclotomic fields, Second edition, Graduate Texts in Mathematics, Springer-Verlag, New Yor, 997. [8] D. G. Zill, M. R. Cullen, Advanced engineering mathematics, Second edition, Jones and Bartlett Publishers, Massachusetts,.