Abstract
Relying on the Hurwitz formula, we find closed-form formulas for the series over sine and cosine functions through the Hurwitz zeta functions, and using them and another summation formula for trigonometric series, we obtain a finite sum for some series over the Riemann zeta functions. We apply these results to the series over Bessel functions, expressing them first as series over the Riemann zeta functions.
Keywords:
Riemann’s zeta function; Hurwitz’s zeta function; gamma function; harmonic numbers; Bessel functions; spherical Bessel functions MSC:
11M35; 33B15; 33E20
1. Preliminaries
Apostol [] (Theorem 12.6, p. 257) derived Hurwitz’s formula
where is the Hurwitz zeta function with and . We can write it as
Theorem 1.
For and , there holds the following formulas:
where .
Proof.
We replace a in Equation (1) first with , and then with . Further, we multiply the first equality by , and the latter by . After subtracting, we obtain
Hence, comparing the corresponding real and imaginary parts, we simultaneously arrive at both formulas in Equation (3). □
For in the first formula of Equation (3), we have
The left-hand series is called the Clausen functions, denoted by .
Similarly, setting in the second formula of Equation (3), we have
and after repeating the preceding procedure, we obtain
The left-hand series is also called the Clausen functions, but denoted by .
Remark 1.
The results (4) and (5) are new, seeing as we have expressed the Clausen functions through the Hurwitz functions, i.e., their first derivative. Because of the similarity regarding the form with some trigonometric series over the sine and cosine functions, there is confusion that these results already exist in some books of tables and integrals. For instance, in [], (p. 726, Section 5.4.2, entries 5. and 6.) the denominator is with no entries, where the denominator in the sine series is . Also, in [], (Section 5.4.2, entries 7. and 8.) the denominator is with no entries, where the denominator in the cosine series is . In [], (Section 5.4.2, entries 1. and 2.), the authors state general formulas for trigonometric series over the sine and cosine functions. Nevertheless, they do not thence derive the cases for the sine series nor for the cosine series. In addition, we have derived more general formulas comprising theirs.
2. Results Related to Series over Zeta Function
Based on the results in the initial section, we shall obtain a finite sum of the series over the zeta functions.
Theorem 2.
For , there holds
where stands for the nth harmonic number.
Proof.
For , we consider the first formula ([], p. 445)
Since we encounter singularities if we set in the first term and the series member for , we must take the limit , , i.e.,
Relying on the L’Hôpital rule, we obtain
We substitute in the right-hand-side series Equation (7) the running index j for k by , and then apply the relation
Theorem 3.
For , there holds
Proof.
For , taking account of the second formula ([], p. 445)
after letting , , it follows
By bringing the fractions in brackets to the same denominator and applying L’Hôpital’s rule, we determine the above limiting value
We are dealing now with a more general series over the Riemann zeta function, reducing it to the series Equations (6) and (9).
Theorem 4.
For and , there holds
Proof.
We make a rational function decomposition of the left-hand side in Equation (11)
with . Further, we have
and determine the constants by applying Heaviside’s method. Knowing that for , it yields
Thus, we can express the left-hand-side series in Equation (11) as follows
In the next step, we shall calculate constants in the representation
where . So, we first multiply the numerator and denominator of the left-hand-side fraction by the missing factors between and , including the latter. Then, we bring the right-hand side fractions to the same denominator, obtaining this way the equality of the numerators
The highest power on the left-hand side and the product at is . There follows . After rearrangements, we have
Regarding both sides as polynomial functions in n of the degree , we take the th derivative. As a result, we have
Hence, we find . Replacing this value in Equation (14), we can determine similarly
Now, the polynomials on both sides are of the degree , and we take the th derivative and obtain
which implies . By repeating this procedure, we obtain
Since
we arrive at the relation
3. Applications to Some Series over Bessel Functions
Bessel functions , , defined by the Swiss mathematician Daniel Bernoulli, then generalized and developed by Friedrich Bessel while studying the dynamics of gravitational systems in the second decade of the 19th century, are canonical solutions of Bessel’s homogenous differential equation
for an arbitrary complex number . The particular solution
is called Bessel’s function of the first kind order []. Another way to represent it is by Poisson’s integral
proved by Poisson [] and Lommel [] to be a solution of Bessel’s homogenous differential equation for ; relying on the summation of trigonometric series, in [], we derived summation formulas for the series
where .
Providing , , the right-hand-side series in Equation (19) truncates because Riemann’s zeta function equals zero if the argument is a negative even integer. So, setting in (19) brings the series in closed form
where .
In the case , we cannot immediately place this on the right-hand side of the relation Equation (19) because one encounters singularities, i.e., in within the numerator of the first term and in the member of the right-hand series for the index .
Theorem 5.
For , there holds
with harmonic numbers and .
Proof.
Because of what is said above, it is necessary to take the limiting value in Equation (19) when
For , all the terms have no singularities if , so it suffices to deal only with the term for ; then, we find
where is Euler–Mascheroni’s constant. Thus, we have
As for the remainder, we transform it by shifting the index, so we introduce the substitution and then revert to k instead of using j, i.e.,
In a special case, setting in Equation (21), we have
Relying on Legendre’s duplication formula for the gamma function ([], p. 35)
where we set , we modify the right-hand side series in Equation (22)
and referring to Equation (9), we have
By differentiating the basic property of Hurwitz’s function
with respect to s and then putting () and (), for the last term of Equation (24), we find
Taking this substitution to the formula Equation (24), we have
4. Series over Spherical Bessel Functions
So, multiplying both sides of Equation (19) by and setting , for , we have
If we place in Equation (28), we obtain a finite sum on the right-hand side
because the series over values truncate for since .
However, we cannot set right away in the right-hand side of Equation (28), since in the numerator of the first term is then . There appears as well for the running index of the right-hand side series, which implies that we encounter in both cases singularities, so we must take a limit. We choose in the right-hand series only term for to evaluate
by bringing to the same denominator and applying the L’Hôpital rule. Afterward, we use Legendre’s duplication formula Equation (23) to rearrange the right-hand side series of (28). Thus, we obtain
Referring to Equation (11) and Equation (9), we evaluate the sum of the right-hand side series. To calculate the harmonic numbers for half-integer indexes, we use the formula Equation (26) with , i.e.
Example 1.
We can obtain the summation formula for the alternating series over spherical Bessel functions. First, considering that there holds
applying Equation (19), we easily find
where we used the relation , and is Dirichlet’s eta function. For , after multiplying this equality by , we have
Further, using the relation
and the equality , where is Dirichlet’s lambda function, we come to the summation formula for the series over a Bessel function containing odd arguments
Setting and multiplying this equality by yields
Finally, replacing Poisson’s integral Equation (18) in the alternating series of Equation (30), we have
Interchanging summation and integration and referring to the formula ([], p. 446)
where is Dirichlet’t beta function, for the right-hand side of Equation (31), we obtain
and then again, interchanging summation and integration, the identity Equation (31) becomes
The last integral is Euler’s beta function
Thus, applying Equation (23), we come to the summation formula of an alternating series over odd arguments Bessel functions:
The beta function vanishes at negative odd integers ([], p. 447). That is why, for , the series over Bessel functions in last formula has a finite sum
Putting and multiplying both sides by , we arrive at the summation formula for the alternating series over spherical Bessel functions
Author Contributions
Both authors have contributed equally. All authors have read and agreed to the published version of the manuscript.
Funding
This research received no external funding.
Data Availability Statement
Data is contained within the article.
Conflicts of Interest
The authors declare no conflicts of interest.
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