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A research on the recursive value and general terms of the analogue Euler zeta function on positive integers
Advances in Difference Equations volume 2013, Article number: 182 (2013)
Our aim is to find the general term of the analogue Euler zeta function in positive integers by using Fourier series. We also figure out the generalized coefficients of Fourier series and investigate some interesting relation in the integers.
Many mathematicians have studied various kinds of an analogue zeta function such as Dirichlet L-function [1–28]. Friedman and Cohen constructed the p-adic analogue for Hurwitz zeta functions . By using multiple Volkenborn integrals, Tangedal and Young defined a p-adic multiple zeta function and a log gamma function [1–18]. Ryoo, Kim and Kim have defined various analogue zeta functions to combine the Euler numbers and Bernoulli numbers [1–12, 17, 18].
Therefore one of the most important and fascinating functions is the zeta function in mathematics [9, 10]. Bernhard Riemann (1826-1866) found something amazing; namely the Riemann zeta function. He recognized the importance of the function onto the entire complex plane ℂ except .
If , where , then
implies that converges uniformly on .
The Riemann zeta function is defined usually by
Some values can be calculated explicitly but , where are still mysterious. The number was demonstrated to be an irrational number by Apery (French mathematician) and can be seen in Hardy, Grosswald, Zhang, Srivastava, and others [20–28]. This Riemann zeta function is the Dirichlet zeta-function, the special case that arises when we take .
In 1837, Lejeune Dirichlet modified the zeta function and he separated the primes into separate categories. The primes depend on the remainder when divided by k. His modified zeta function is of the form
where is a special kind of the function . Any function of the form is known as a Dirichlet L-series where s is a real number greater than 1 and χ is called as Dirichlet character.
Mathematicians have studied extensively the Riemann zeta function and the Dirichlet zeta function because these functions play an important role in physics, complex analysis and number theory etc. They also recognized that the discovery of these zeta functions dates back to Euler.
Leonhard Euler (1707-1783) defined the zeta function for any real number greater than 1 by the infinite sum. After Euler defined this function, he showed that it had a deep and profound connection with the pattern of the primes. He also calculated , and has been researched, proved by mathematicians . The values of the Riemann zeta function were computed by the Euler zeta function at even positive integers. The analogue Euler zeta function replaces the Euler zeta function by mathematician’s research. The Euler zeta function was originally constructed by Kim (see ) and Kim gave the values of the Euler zeta function as positive integers (see [, Theorem 3.1]). Kim, Choi, and Kim researched to combine the Euler numbers and Bernoulli numbers in order to get values or a generalized term .
The Euler zeta function is defined as follows.
Definition 1.1 For , and ,
From Definition 1.1, we define the analogue Euler zeta function as follows.
Definition 1.2 Let , and .
We easily note that .
In this paper, we find out the generalized coefficients of Fourier series and investigate some interesting relations in the positive integers. We also investigate values and a generalized term of the analogue Euler zeta function in the same way by using the Fourier series.
The Fourier series can be expressed as summation between sine series and cosine series instead of complicated functions.
If the function has period 2p, then the Fourier series of is
where coefficients of the Fourier series , , and are defined by the integrals
A special instance of the Fourier series is the cosine series. If is initially defined over the interval , then it can be extended to and then extended periodically with period 2p. So, the cosine series of the Fourier series on is defined by
The sine series is a special instance of the Fourier series. Let . Then can be extended to . The Fourier series for this odd, periodic function reduces to the sine series in the form
because , each .
We denote that , , and when .
The paper is organized as follows. In Section 2, we construct generalized coefficients of sine series and cosine series in the positive integers and prove them. We also study some interesting relations about sine series and cosine series in the positive integers. In Section 3, applying these ideas, generalized coefficients will be used to obtain the main results of this paper. We also find the general term of the analogue Euler zeta function.
2 The coefficient’s rule of cosine series and sine series
In this section, we construct the coefficient’s rule of cosine series and sine series. We access some relations about the coefficient sine series and cosine series.
The cosine series for is given by the following theorem.
Theorem 2.1 Let for and l be a positive even integer.
Proof We shall prove Theorem 2.1 using mathematical induction. We assume that for and l is a positive even integer.
Clearly, and hold when as follows.
Suppose that is true for and in the positive even integers. That is,
Consider the case (m: even).
By the above assumption, we get
That is, and hold for (m: even) if it holds for . Thus, we complete the proof of the theorem. □
Remarks Let for . Then we have the following equation:
Let for . From Theorem 2.1, we get
From (2.2), we get an interesting rule of coefficient .
Let for . The cosine series of is the following equation.
From (2.3), we are able to express the rule of coefficient by using the matrix
Let for . Then the cosine series of is the following equation:
From the above equation, we are able to represent the rule of coefficient by using the matrix
By using the matrix, we can display the coefficient of cosine series in Theorem 2.1 as follows.
Corollary 2.2 Let l be a positive even integer. Then one has
From now on, we will see the coefficients of the sine series. The sine series is given by the following theorem.
Theorem 2.3 Let for and l be a positive odd integer.
Proof (By mathematical induction) Where , says that
which is true.
Now fix that (m: odd) is true for and suppose that holds, that is,
By our hypothesis on , we see that
Therefore holds for (m: odd).
Thus, we conclude the proof of the theorem by the principle of mathematical induction. □
Let for . Then we have the following equation:
Let for . From Theorem 2.3, we get
From (2.5), we also get an interesting rule of coefficient .
Let for . The sine series of is the following equation:
From (2.6), we are able to express the rule of coefficient by using the matrix
Let for . Then the sine series of is the following equation:
From the above equation, we are able to represent the rule of coefficient by using the matrix
By using the matrix, we can arrange the coefficient of sine series in Theorem 2.3 as follows.
Corollary 2.4 Let l be any positive odd integer. Then we get
From Theorem 2.1 and Theorem 2.3, we get the relation of coefficients between and .
Theorem 2.5 Let : the coefficient of cosine series, : the coefficient of sine series, and .
Proof Take () in Theorem 2.3. Then we easily see that
We also use () from Theorem 2.1. Then we obtain the following equation:
If l is an even integer, then .
Example 2.6 We state the relation of and . Let : the coefficient of cosine series, : the coefficient of sine series, and . By using Theorem 2.5, we derive that
3 The analogue Euler zeta function in the integers
In this section, we get the value and generalized term of the analogue Euler zeta function. We derive by using the coefficient of cosine series and obtain using the coefficient of sine series.
Using , Theorem 2.1 can be written as
Taking in (3.1), we easily find the following equation:
since is if and 0 if with . Therefore, we have the results as follows:
From (3.3), we get some analogue Euler zeta functions as follows.
Hence, we find out the following generalized term , when m is a positive even integer.
where recursively we define (for an even positive integer m)
From the above, we obtain the following theorem.
Theorem 3.1 Let m be any positive even integer. Then we have
By observing Theorem 3.1, we can easily understand the relation between the Euler zeta function and the analogue Euler zeta function.
Example 3.2 From Theorem 3.1, in case of we derive that
By Theorem 2.3, we note that
Let , then we have the following equation:
since is if and 0 if with .
Hence, we have equation (3.6) from (3.5).
From (3.6) with , we obtain the following.
From the above equations, we have the generalized term.
where recursively we define (for an odd positive integer m)
Thus, we have Theorem 3.2.
Theorem 3.3 Let m be a positive odd integer. Then one has
Example 3.4 By using Theorem 3.3, in case of we have
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The authors express their gratitude to the referee for his/her valuable comments. We also thank T. Kim and A. Sankaranarayanan for their great support and encouragement. They kindly offered invaluable lecture and advice about the paper. This work was supported by NRF (National Research Foundation of Korea) Grant funded by the Korean Government (NRF-2013-Fostering Core Leaders of the Future Basic Science Program).
The authors declare that they have no competing interests.
Both authors contributed equally to the manuscript and read and approved the final manuscript.
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Kang, J., Ryoo, C. A research on the recursive value and general terms of the analogue Euler zeta function on positive integers. Adv Differ Equ 2013, 182 (2013). https://doi.org/10.1186/1687-1847-2013-182
- Fourier series
- cosine series
- sine series
- analogue Euler zeta function