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Junesang Choi, "Rapidly Converging Series for from Fourier Series", Abstract and Applied Analysis, vol. 2014, Article ID 457620, 9 pages, 2014. https://doi.org/10.1155/2014/457620
Rapidly Converging Series for from Fourier Series
Ever since Euler first evaluated and , numerous interesting solutions of the problem of evaluating the have appeared in the mathematical literature. Until now no simple formula analogous to the evaluation of is known for or even for any special case such as . Instead, various rapidly converging series for have been developed by many authors. Here, using Fourier series, we aim mainly at presenting a recurrence formula for rapidly converging series for . In addition, using Fourier series and recalling some indefinite integral formulas, we also give recurrence formulas for evaluations of and , which have been treated in earlier works.
1. Introduction and Preliminaries
The Riemann zeta function is defined by (see, e.g., [1, p. 164]) The Riemann zeta function in (1) plays a central role in the applications of complex analysis to number theory. The number-theoretic properties of are exhibited by the following result known as Euler's formula, which gives a relationship between the set of primes and the set of positive integers: where the product is taken over all primes.
The solution of the so-called Basler problem (cf., e.g., , [3, p. xxii], [4, p.66], [5, pp. 197-198], and [6, p. 364]) was first found in 1735 by Euler (1707–1783) , although Jakob Bernoulli (1654–1705) and Johann Bernoulli (1667–1748) did their utmost to sum the series in (3). The former of these Bernoulli brothers did not live to see the solution of the problem, and the solution became known to the latter soon after Euler found it (see, for details, Knopp [8, p.238]). Five years later in 1740, Euler (see ; see also [10, pp. 137–153]) succeeded in evaluating all of ): where () are the th Bernoulli numbers defined by the following generating function (see, e.g., [1, p. 81]): The following recursion formula can be used for computing Bernoulli numbers. Ever since Euler first evaluated and , numerous interesting solutions of the problem of evaluating the have appeared in the mathematical literature. Even though there were certain earlier works which gave a rather long list of papers and books together with some useful comments on the methods of evaluation of and (see, e.g., [5, 11, 12]), the reader may be referred to the very recent work  which contains an extensive literature of as many as more than 70 papers.
The eta function or the alternating zeta function is defined by Then it is easy to have the following relation between and :
The -function is defined by
Remark 1. The -function (see [15, p. 329]) has been denoted in several ways, such as (see ), (see [15, p. 329]; see also [1, p. 404]), (see [16, p. 125] and ), (see ), (see [19, p. 190]; see also [15, p. 332]), and (see [20, p. 375]). Williams [14, p. 22, Theorem II] gave an interesting companion of the result (7) in the following form: which appears erroneously in Hansen [21, p. 357, Entry (54.7.1)]. Since is the well-known Gregory series for (with being the familiar Catalan constant ), by setting in (11), we immediately obtain
Until now no simple formula analogous to (4) is known for or even for any special case such as . It is not even known whether is rational or irrational, except that the irrationality of was proved by Apéry . But it is known that there are infinitely many which are irrational (see [23, 24]). On the other hand, various rapidly converging series for have been developed by many authors (see, e.g., [25, 26]; see also [1, Chapter 4] and the references cited in the chapter). Very recently, Choi and Chen  gave a double inequality approximating by a more rapidly convergent series. Here, using Fourier series, we aim mainly at presenting a recurrence formula for rapidly converging series for . In addition, using Fourier series, we also give recurrence formulas for evaluations of and .
2. Evaluation of from Fourier Series
Here we present a recurrence formula for evaluation of the given in (10) by using Fourier series. To do this, we choose the odd -periodic function given by which is seen to be continuous and piecewise differentiable on the set of real numbers . Now we can get the following Fourier series expansion of : where
Theorem 2. The following recurrence formula for evaluation of holds. For , where the empty sum is (as usual) understood to be nil throughout this paper.
For small values of , we have
Remark 3. In order to get the evaluation of and a rapidly converging series representation of from Fourier series, instead of using the periodic version of which is not continuous, Scheufens  made a good choice of the odd -periodic function given by which is now continuous and piecewise differentiable. Here we use the function in (15) which is a natural modification of the Scheufens chosen function (23) to give the results in this and the next sections.
Chen  used the even -periodic function on to get a recurrence formula for . Yue and Williams  used residue calculus to derive a recurrence formula for . Butzer and Hauss  presented diverse single and multiple integral representations of .
3. A Recurrence Formula for a Rapidly Converging Series for
We begin by recalling some elementary known or easily derivable formulas for the binomial coefficients as in the lemma given below.
Lemma 4. Each of the following formulas holds: where denotes the set of complex numbers, where is defined, for , by
Lemma 5. For each and , one has , where, for convenience,
Proof. We proceed to prove by induction on . We can give a direct evaluation to check the first three ones:
Now assume that for some . Then we have to show that By using the induction hypothesis, we find In view of (31) and (32), it is enough to show that where, for convenience, with
We first try to evaluate . We find Let us consider the following partial fraction: Using (27), we have Applying the last two identities to the last expression of , we can separate into six polynomials as follows: where Choosing to use some identities in Lemma 4, we can evaluate as follows: We thus have
Similarly we find where Similarly as in evaluating , we have We thus obtain Now it is easy to see that This completes the proof of (33) and so does Lemma 5.
Theorem 6. One has a recurrence formula for a rapidly converging series for . For ,
Proof. We find from (15), (16), and (20) that, for and ,
Here we choose a method where the sine function disappears by using the following well-known result:
The series for obtained by dividing (48) by converges uniformly on . Indeed, the value of at can be considered as and then one may use the Weierstrass -test. We therefore apply termwise integration to the resulting series and use (49) to get
where is the eta function given in (8). Now using the relationship (9) with gives
Since is -periodic, we have where, for convenience,
By using the binomial formula and integrating the resulting identity, we get where
Applying the following Maclaurin series to , in view of Lemma 5, we can show that We therefore have
Finally, setting the last expression of in (52) and considering (51) yield our desired identity (47). This completes the proof of Theorem 6.
The special case of (47) when yields which, upon using (see, e.g., [1, p. 165, (10)]), can be expressed in a more compact form: The formula (59) or (60) has already been presented (cf., e.g., [29, p. 31] and [30, p. 837]).
Remark 7. Since , using the th partial sum of the infinite series in (59) or (60), we can compute with an error satisfying
Using the th partial sum in (60) we have an error bound , and approximately the value . For comparison, using the th partial sum of , we can compute with an error satisfying
from which we get the error bound . Scheufens [29, p. 31] estimated the error bounds and . So it is easy to see that the original series of converges very slowly while the series representation (59) or (60) of converges very rapidly.
Srivastava and Choi [1, Chapter 3] presented a rather extensive collection of closed-form sums of series involving the zeta functions such as (59) or (60), together with an interesting historical introduction. In fact, the formula (47) may be obtained in a totally different way (see, e.g., [1, p. 259, (71)]).
4. Evaluation of from Fourier Series
There have been earlier works (see, e.g., [17, 29, 31–34]) in which the authors evaluated by using Fourier series. Here, for completeness, we also do the same thing. Yet we may very carefully emphasize that, when the involved coefficient in Fourier series is computed, its computation becomes a little easier by using a known indefinite integral formula.
For , let be the even -periodic function given by , . Since is continuous and piecewise differentiable, we have where Evaluation of in (65). Use a known indefinite integral formula (see, e.g., [28, p. 211, Entry 2.633(2)]) to get the following equation: For convenience, let the right-hand side of (66) be denoted by . Then we have By considering the following relation we have We finally get
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
This research was supported by the Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology of the Republic of Korea (2010-0011005).
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Copyright © 2014 Junesang Choi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.