Fourier series of sums of products of ordered Bell and poly-bernoulli functions

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1 Kim et al. Journal of Inequalities and Applications 27 27:84 DOI.86/s R E S E A R C H Open Access Fourier series of sums of products of ordered ell and poly-ernoulli functions Taekyun Kim,2,DaeSanKim 3,DmitryVDolgy 4 and Jin-Woo Park 5* * Correspondence: a47@knu.ac.kr 5 Department of Mathematics Education, Daegu University, Gyeongsan-si, Gyeongsangbuk-do 72-74, Republic of Korea Full list of author information is available at the end of the article Abstract In this paper, we study three types of sums of products of ordered ell and poly-ernoulli functions and derive their Fourier series expansion. In addition, we express those functions in terms of ernoulli functions. MSC: 68; 83; 42A6 Keywords: Fourier series; ordered ell function; poly-ernoulli function Introduction The ordered ell polynomials b m x are defined by the generating function 2 e t ext b m x tm m!. m Thus they form an Appell sequence. For x,b m b m, m are called ordered ell numbers which have been studied in various counting problems in number theory and enumerative combinatorics see [, 4, 5, 6, 7, 9]. The ordered ell numbers are all positive integers, as we can see, for example, from n m b m n!s 2 m, n m. 2 n+ n n The first few ordered ell polynomials are as follows: b x, b xx +, b 2 xx 2 +2x +3, b 3 xx 3 +3x 2 +9x +3, b 4 xx 4 +4x 3 +8x 2 +52x +75, b 5 xx 5 +5x 4 +3x 3 +3x x +54. From, we can derive d dx b mxmb m x m, b m x ++2b m xx m m. The Authors 27. This article is distributed under the terms of the Creative Commons Attribution 4. International License which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original authors and the source, provide a link to the Creative Commons license, and indicate if changes were made.

2 Kim etal. Journal of Inequalities and Applications 27 27:84 Page 2 of 7 From these, in turn, we have b m + 2b m δ m, m, b m x dx bm+ m m+ m + m + b m+. For any integer r,thepoly-ernoulli polynomials of index r r m xaregivenbythegenerating function see [2, 3, 7, 2, 3, 2] Li r e t e t e xt m r m x tm m!, where Li r x m xm m r is the polylogarithmic function for r and a rational function for r. We note here that d Lir+ x dx x Li rx. Also,weneedthefollowingforlateruse. d dx r m xmr m x m, m x mx, r x, m xxm, m δ m,, r+ m r+ m r m m, r m x dx r m+ r m+ m + m + r m. Here the ernoulli polynomials m x are given by the generating function t e t ext m x tm m!. m For any real number x,we let x x [, denote the fractional part of x. Finally, we recall the following facts about ernoulli functions m : a for m 2, m m! n n e x m ;

3 Kim etal. Journal of Inequalities and Applications 27 27:84 Page 3 of 7 b for m, n n e x, for x / Z,, for x Z. Here we will study three types of sums of products of ordered ell and poly-ernoulli functions and derive their Fourier series expansion. In addition, we will express those functions in terms of ernoulli functions. α m m b k r+, m ; 2 β m m k!! b k r+, m ; 3 γ m m k k b k r+, m 2. For elementary facts about Fourier analysis, the reader may refer to any book for example, see [8, 2]. As to γ m, we note that the next polynomial identity follows immediately from Theorems 4. and 4.2, which is in turn derived from the Fourier series expansion of γ m : m k km k b kx r+ x m s m s m s+ + H m H m s m s + r m s + b m s+ s x, where H l l j j are the harmonic numbers and l l kl k b k r+ k l l k +2 k kl k b k r l k, with. The polynomial identities can be derived also for the functions α m andβ m from Theorems 2. and 2.2, andtheorems3. and 3.2, respectively. We refer the reader to [6,, 4, 5] for the recent papers on related works. 2 Fourier series of functions of the first type Let α m x b k x r+ x, where r, m are integers with m. Then we will study the function α m m b k r+ m, defined on R which is periodic with period.

4 Kim etal. Journal of Inequalities and Applications 27 27:84 Page 4 of 7 The Fourier series of α m is where n A m n A m n ex, α m e x dx α m xe x dx. efore proceeding further, we observe the following m α m x { kbk x r+ x+m kb k r+ kb k x r+ k x+ m m k +b k x r+ m +α m x. x} m kb k x r+ x+ m x m kb k x r+ x Thus α m+x m+2 α m x, and so α mx dx m+2 α m+ α m+. For m, we put m α m α m m b k r+ b k r+ m 2b k δ k, r+ + r +2bm δ m, b k r+ b k r+ m 2 b k r+ k +2 m b k r +2 m r+ m r b k r r m. m + b m m Thus, α m α m m and α mx dx m+2 m+. Now, we want to determine the Fourier coefficients A m n. Case : n. b k r+ A m n α m xe x dx [ αm xe x] + α m xe x dx

5 Kim etal. Journal of Inequalities and Applications 27 27:84 Page 5 of 7 m + m + Am m + α m xe x dx αm α m n m m Am 2 n m m Am 2 n 2 j m + m m m A n j j m + j j m j+ m + j j m j+ j j m j+, wherewenotethata n e x dx. Case 2: n. A m α m x dx m+. m α m, m is piecewise C.Moreover,α m is continuous for those positive integers m with m and discontinuous with jump discontinuities at integers for those positive integers m with m. Assume first that m is a positive integer with m.thenα m α m. Hence α m is piecewise C and continuous. Thus the Fourier series of α m converges uniformly to α m, and α m m+ + n n m+ + j j j m j+ m j+ j! j j m+ + j j2, for x / Z, + m, for x Z. Now, we can state our first theorem. Theorem 2. For each positive integer l, we let l l b k r+ k l l k +2 b k r l k r l. n n m j+ j e x e x j Assume that m for a positive integer m. Then we have the following

6 Kim etal. Journal of Inequalities and Applications 27 27:84 Page 6 of 7 a m b k r+ has the Fourier series expansion b k r+ m+ + n n j j m j+ e x, j b for all x R, where the convergence is uniform. b k r+ m+ + for all x R, where j is the ernoulli function. j j2 m j+ j, Assume next that m for a positive integer m. Thenα m α m. So α m is piecewise C and discontinuous with jump discontinuities at integers. The Fourier series of α m convergespointwisetoα m, for x / Z,andconvergesto αm + α m α m m, for x Z. Now, we can state our second theorem. Theorem 2.2 For each positive integer l, we let l l b k r+ k l l k +2 b k r l k r l. Assume that m for a positive integer m. Then we have the following. a b m+ + n n j j j m j+ m b k r+, for x / Z, m b k r+ + 2 m, for x Z; m+ + m j+ j j j b k r+, for x / Z; e x

7 Kim etal. Journal of Inequalities and Applications 27 27:84 Page 7 of 7 m+ + m j+ j j j2 b k r+ + 2 m, for x Z. 3 Fourier series of functions of the second type Let β m x m k!! b kx r+ x, where r, m are integers with m. Then we will investigate the function β m m k!m k! b k r+, defined on R, which is periodic with period. The Fourier series of β m is where n m n m n ex, β m e x dx β m xe x dx. eforeproceedingfurther,wenotethefollowing. m β m x { k m k k!m k! b k x r+ x+ m k k!m k! b kx r+ x } m k!m k! b k x r+ x+ k!m k! b kx r+ x m k!m k! b kx r+ x+ 2β m x. k!m k! b kx r+ x Thus βm+ x β m x, and 2 β m x dx 2 βm+ β m+. For m, we put m β m β m k!m k! b k r+ m k!m k! b k r+

8 Kim etal. Journal of Inequalities and Applications 27 27:84 Page 8 of 7 m k!m k! 2b k δ k, r+ + r + m! 2b m δ m, m 2 m k!m k! b k r+ +2 m! r+ m m! r k k!m k! b k r+ m + m! b m m k!m k! b k r k!m k! b k r+ m k!m k! b k r+ +2 k!m k! b k r m! r m. Hence β m β m m, and β m x dx 2 m+. We now would like to determine the Fourier coefficients m n. Case : n. m n β m xe x dx [ βm xe x] + βm β m + 2 β m xe x dx β m xe x dx 2 m n m 2 2 m 2 n m m 2 2 m 2 n 2 j 2 j m m n j 2 j j m j+. 2 j j m j+ 2 j j m j+ Case 2: n. m β m x dx 2 m+.

9 Kim etal. Journal of Inequalities and Applications 27 27:84 Page 9 of 7 β m, m is piecewise C.Moreover,β m is continuous for those positive integers m with m and discontinuous with jump discontinuities at integers for those positive integers m with m. Assume first that m is a positive integer with m.thenβ m β m. Thus β m is piecewise C and continuous. Hence the Fourier series of β m converges uniformly to β m, and β m 2 m+ + 2 m+ + 2 m+ + n n j j2 2 j j! 2 j j! j 2 j j m j+ e x m j+ j! m j+ j, for x / Z, + m, for x Z. n n e x j Now, we can state our first result. Theorem 3. For each positive integer l, we let l l k k!l k! b k r+ l l k +2 k!l k! b k r l k l! r l. Assume that m for a positive integer m. Then we have the following. a m k!! b k r+ has the Fourier series expansion k!m k! b k r+ 2 m+ + n n j 2 j j m j+ e x, b for all x R, where the convergence is uniform. k!m k! b k r+ 2 m+ + for all x R, where j is the ernoulli function. j2 2 j j! m j+ j, Assume next that m for a positive integer m. Thenβ m β m. So β m is piecewise C and discontinuous with jump discontinuities at integers. The Fourier series

10 Kim etal.journal of Inequalities and Applications 27 27:84 Page of 7 of β m convergespointwisetoβ m, for x / Z,andconvergesto βm + β m β m m, for x Z. Now, we can state our second theorem. Theorem 3.2 For each positive integer l, we let l l k k!l k! b k r+ l l k +2 k!l k! b k r l k l! r l. Assume that m for a positive integer m. Then we have the following. a b 2 m+ + 2 m+ + n n j m k!! b k r+ m 2 m+ + j 2 j j m j+ e x, for x / Z, k!! b k r+ + 2 m, for x Z. 2 j j! m j+ j k!m k! b k r+, for x / Z; j2 2 j j! m j+ j k!m k! b k r+ + 2 m, for x Z. 4 Fourier series of functions of the third type Let m γ m x km k b kx r+ x, k where r, m are integers with m 2. Then we will consider the function m γ m km k b k r+, k defined on R, which is periodic with period.

11 Kim etal.journal of Inequalities and Applications 27 27:84 Page of 7 The Fourier series of γ m is where n C m n C m n ex, γ m e x dx γ m xe x dx. efore proceeding further, we need to observe the following. m γ m x k m 2 m 2 k m m k b k x r+ x+ k m m k b kx r+ x+ m k + k m γ m x+ k b kx r+ x k k b kx r+ x b k x r+ x+ m r+ m x+ m b m x m r+ m x+ m b m x. From this, we see that γ m+ x m mm + r+ m+ x mm + b m+x γ m x, and γ m x dx m m [ γ m+ x γ m+ γ m ] mm + r+ m+ x mm + b m+x mm + r+ m+ r+ m+ bm+ b m+ mm + γ m+ γ m+ m mm + r m mm + b m+. For m 2, we let m γ m γ m m bk r+ km k b k r+ k

12 Kim etal.journal of Inequalities and Applications 27 27:84 Page 2 of 7 Then m 2bk δ k, r+ km k + r bk r+ k m k m km k b k r+ +2 k km k b k r. γ m γ m m, and γ m x dx m m+ mm + r m mm + b m+. Now, we would like to determine the Fourier coefficients C m n. Case : n. For this computation, we need to know the following: where l xe x l l k k r dx k l k, forn, l+ r l, forn, b l xe x l l k k b dx k l k+, forn, l+ b l+, forn, r+ C m n γ m xe x dx [ γm xe x] + γm γ m + m γ m x+ m Cm n m + m + m m Cm n m b m xe x dx γ m xe x dx m r+ m x+ m m m b m x r+ m xe x dx m m, e x dx m γ m γ m m k m m k m km k b k r+ +2 m k r k k, m m km k b k r, k m k k b.

13 Kim etal.journal of Inequalities and Applications 27 27:84 Page 3 of 7 C n m m Cm n m m m 2 Cm 2 n m m m 2 2 Cm 2 n m m 2 j m m m m m j j m j+ m 2 m m m m 2 m 2 j m j j m j m j+ 2 j m j j m j m j+ m j m j m j m j+ j m j j m j m m j+ j m j j m j m j+. We note here that m j m j j m j m j+ m j j m j m j j m j k m j k m j k b k m j k+ m m j+k 2 j+k m j b m j k+ m j j m j m j k m m j s2 s m sj+ m s 2 s m j+k 2 b j+k m j k+ m s 2 b s m s+ b m s+ s j m j m s 2 s b m s+ H m H m s m s H m H m s b s m s+. m s + s Putting everything together, we get C m n m s { m s s m s+ + H m H m s m s + r m s + b m s+ }.

14 Kim etal.journal of Inequalities and Applications 27 27:84 Page 4 of 7 Case 2: n. C m m γ m x dx m+ mm + r m mm + b m+. γ m, m 2ispiecewiseC.Moreover,γ m is continuous for those integers m 2 with m and discontinuous with jump discontinuities at integers for those integers m 2with m. Assume first that m.thenγ m γ m. Thus γ m ispiecewisec and continuous. Hence the Fourier series of γ m converges uniformly to γ m, and γ m m+ m mm + r m mm + b m+ { + m n n + m m m+ s m s s m s s m s+ + H m H m s m s + mm + r m mm + b m+ m s+ + H m H m s r m s m s + + b m s+ s! m+ m mm + r m mm + b m+ + m m s+ + H m H m s m s m s + s2, for x / Z, + m, for x Z m s s m s m s+ + H m H m s m s +, for x / Z, + m, for x Z. r m s + b } m s+ e x n n r m s + b m s+ s r m s + b m s+ s e x s Now, we can state our first result. Theorem 4. For each integer l 2, we let l l kl k b k r+ k l l k +2 k kl k b k r l k, with.assume that m for an integer m 2. Then we have the following.

15 Kim etal.journal of Inequalities and Applications 27 27:84 Page 5 of 7 a m k k b k r+ has Fourier series expansion m k km k b k m + r+ m+ mm + r m mm + b m+ { m n n s m s s m s+ + H m H m s m s + r m s + b m s+ } e x, b for all x R, where the convergence is uniform. m k km k b k m s s m s r+ m s+ + H m H m s m s + r m s + b m s+ s, for all x R, where s is the ernoulli function. Assume next that m is an integer 2with m.thenγ m γ m. Hence γ m is piecewise C and discontinuous with jump discontinuities at integers. Then the Fourier series of γ m convergespointwisetoγ m, for x / Z,andconvergesto γm + γ m γ m m m km k b k r+ + 2 m, k for x Z. Now, we can state our second result. Theorem 4.2 For each integer l 2, let l l kl k b k r+ k l l k +2 k kl k b k r l k, with.assume that m for an integer m 2. Then we have the following. a m + m+ mm + r m mm + b m+ { m n n s m s s m s+ + H m H m s m s + r m s + b } m s+ e x

16 Kim etal.journal of Inequalities and Applications 27 27:84 Page 6 of 7 b m m m k m k s k k b k r+, for x / Z, k b k r+ + 2 m, for x Z. m s m s+ + H m H m s m s + m km k b k r+, for x / Z; s s k m s m s+ + H m H m s m s + m km k b k r+ + 2 m, for x Z. 5 Results and discussion r m s + b m s+ s r m s + b m s+ s In this paper, we study three types of sums of products of ordered ell and poly-ernoulli functions and derive their Fourier series expansion. In addition, we express those functions in terms of ernoulli functions. The Fourier series expansion of the ordered ell and poly-ernoulli functions are useful in computing the special values of the poly-zeta and multiple zeta function. For details, one is referred to [3, 7 8]. It is expected that the Fourier series of the ordered ell functions will find some applications in connection with a certain generalization of the Euler zeta function and the higher-order generalized Frobenius-Euler numbers and polynomials. 6 Conclusion In this paper, we considered the Fourier series expansion of the ordered ell and poly- ernoulli functions which are obtained by extending by periodicity of period the ordered ell and poly-ernoulli polynomials on [,. The Fourier series are explicitly determined. 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. Author details Department of Mathematics, College of Science, Tianjin Polytechnic University, Tianjin, 36, China. 2 Department of Mathematics, Kwangwoon University, Seoul, 39-7, Republic of Korea. 3 Department of Mathematics, Sogang University, Seoul, 2-742, Republic of Korea. 4 Hanrimwon, Kwangwoon University, Seoul, 39-7, Republic of Korea. 5 Department of Mathematics Education, Daegu University, Gyeongsan-si, Gyeongsangbuk-do 72-74, Republic of Korea. Acknowledgements The first author has been appointed a chair professor at Tianjin Polytechnic University by Tianjin City in China from August 25 to August 29. Publisher s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Received: 26 February 27 Accepted: 7 April 27

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