Series of sums of products of higher-order Bernoulli functions

It is shown in a previous work that Faber-Pandharipande-Zagier’s and Miki’s identities can be derived from a polynomial identity, which in turn follows from the Fourier series expansion of sums of products of Bernoulli functions. Motivated by and generalizing this, we consider three types of functions given by sums of products of higher-order Bernoulli functions and derive their Fourier series expansions. Moreover, we express each of them in terms of Bernoulli functions.


Introduction
Let r be a nonnegative integer. Then the Bernoulli polynomials B (r) n (x) of order r are given by the generating function (see [ (.) In this paper, we will study the following three types of sums of products of higher-order Bernoulli functions and find Fourier series expansions for them. Moreover, we will express them in terms of Bernoulli functions. Let r, s be positive integers. ( . For elementary facts about Fourier analysis, the reader may refer, for example, to [-]. As to γ m ( x ), we note that the polynomial identity (.) follows immediately from the Fourier series expansion of γ m ( x ) in Theorems . and .: where, for each integer l ≥ , and H m = m j=  j are the harmonic numbers. It is remarkable that the famous Faber-Pandharipande-Zagier identity (see [, ]) and the Miki identity (see [-]) can be easily derived from (.) and (.), with r = s = . Below, we will give an outline for this and thus this may be viewed as our main motivation for the present study.
Indeed, from (.) and (.), with r = s = , we get Simple modification of (.) yields Letting x =  in (.) gives a slightly different version of the well-known Miki identity (see []): which is the Faber-Pandharipande-Zagier identity (see [] . Then we will consider the function defined on R, which is periodic with period .
To continue our discussion, we need to observe the following: From this, we obtain We now would like to determine the Fourier coefficients A (m) n . Case : n = .
from which by induction on m, we can easily derive that We are now ready to state our first result.
Theorem . For each positive integer l, let Assume that m = , for a positive integer m. Then we have the following.
for all x ∈ R, where the convergence is uniform.
for all x in R.
Assume next that m = , for a positive integer m.
is piecewise C ∞ , and discontinuous with jump discontinuities at integers.
Now, we are ready to state our second result.
Theorem . For each positive integer l, let Assume that m = , for a positive integer m. Then we have the following.
. Then we will study the function Before proceeding, we need to observe the following: From this, we get We now want to determine the Fourier coefficients B (m) n .
From this, we easily get the following result by induction on m: (.) Now, we can state our first result.

Theorem . For each positive integer l, let
Assume that m = , for a positive integer m. Then we have the following.
for all x ∈ R, where the convergence is uniform.
for all x ∈ R.
Assume next that m = , for a positive integer m. Then β m () = β m (). Hence β m ( x ) is piecewise C ∞ , and discontinuous with jump discontinuities at integers. Then the Fourier series of β m ( x ) converges pointwise to β m ( x ), for x / ∈ Z, and converges to for x ∈ Z. Now, we can state our second result.

Theorem . For each positive integer l, let
Assume that m = , for a positive integer m. Then we have the following: . Then we will investigate the function To proceed, we need to observe the following: From this, we easily obtain Let  = , and for m ≥ , we let Then evidently we have We now would like to determine the Fourier coefficient C (m) n .
f o r n = . (.) Thus we have shown that from which, by induction on m, we can show that Here we note that Finally, we get the following expression of C (m) n , for n = : In addition, γ m ( x ) is continuous for those integers m ≥  with m = , and discontinuous with jump discontinuities at integers for those integer m ≥  with m = . Assume first that m = , for an integer m ≥ . Then γ m () = γ m (). Hence γ m ( x ) is piecewise C ∞ , and continuous. Thus the Fourier series of γ m ( x ) converges uniformly to γ m ( x ), and (.) Now, we are ready to state our first result.
with  = . Assume that m = , for an integer m ≥ . Then we have the following.
for all x ∈ R, where the convergence is uniform.
for all x ∈ R.
Assume next that m = , for an integers m ≥ . Then γ m () = γ m (). 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, and converges to for x ∈ Z. Now, we can state our second result.
with  = . Assume that m = , for an integer m ≥ . Then we have the following: (.)

Results and discussion
It is shown in a previous work that Faber-Pandharipande-Zagier's and Miki's identities can be derived from a polynomial identity, which in turn follows from the Fourier series expansion of sums of products of Bernoulli functions. Motivated by and generalizing this, we consider three types of functions given by sums of products of higher-order Bernoulli functions and we obtain some new identities arising from Fourier series expansions associated with sums of products of higher-order Bernoulli functions. Moreover, we will express each of them in terms of Bernoulli functions. The Fourier series expansion of the sums of products of higher-order Bernoulli functions are useful in computing the special values of the zeta and multiple zeta function. It is expected that the Fourier series of the sums of products of higher-order Bernoulli functions will find some applications in connection with a certain zeta function and the higher-order Bernoulli numbers.

Conclusion
In this paper, we considered the Fourier series expansion of the sums of products of higherorder Bernoulli functions which are obtained by extending by periodicity of period the Bernoulli polynomials on [, ). The Fourier series are explicitly determined.