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Certain integral representations of Stieltjes constants
Journal of Inequalities and Applicationsvolume 2013, Article number: 532 (2013)
A remarkably large number of integral formulas for the Euler-Mascheroni constant γ have been presented. The Stieltjes constants (or generalized Euler-Mascheroni constants) and , which arise from the coefficients of the Laurent series expansion of the Riemann zeta function at , have been investigated in various ways, especially for their integral representations. Here we aim at presenting certain integral representations for by choosing to use three known integral representations for the Riemann zeta function . Our method used here is similar to those in some earlier works, but our results seem a little simpler. Some relevant connections of some special cases of our results presented here with those in earlier works are also pointed out.
MSC:11M06, 11M35, 11Y60, 33B15.
1 Introduction and preliminaries
The Riemann zeta function is defined by (see, e.g., [, Section 2.3])
which is an obvious special case of the Hurwitz (or generalized) zeta function defined by
where ℂ and denote the sets of complex numbers and nonpositive integers, respectively. Both the Riemann zeta function and the Hurwitz zeta function can be continued meromorphically to the whole complex s-plane, except for a simple pole only at , with their respective residue 1, in many different ways. The Stieltjes constants for , , arise from the following Laurent expansion of the Riemann zeta function about (see, e.g., [, pp.166-169], [, p.255] and [, p.165]):
The Stieltjes constants are named after Thomas Jan Stieltjes and often referred to as generalized Euler-Mascheroni constants. Liang and Todd  computed numerical approximations of the first 20 Stieltjes constants in 1972. In 1985, using contour integration, Ainsworth and Howell  showed that
By using binomial theorem, we have
where, for convenience and simplicity,
From (1.6) and (1.8), we obtain a more explicit integral representation for the Stieltjes constants :
where and are given in (1.8). Similarly, we have
where, for convenience and simplicity,
From (1.6) and (1.10), we get a more explicit integral representation for the Stieltjes constants :
We recall the polygamma functions () defined by
where denotes the psi (or digamma) function defined by
where are the complete Bell polynomials defined by and
the sum being taken over all partitions of n, i.e., over all sets of such that
A remarkably large number of integral formulas for the Euler-Mascheroni constant γ have been presented (see, e.g., [9, 10], and [, Section 1.2]). The Stieltjes constants () have been investigated in various ways, especially for their integral representations (see, e.g., [4–8]; see also [, Section 2.21] and the references cited therein). Here we aim at presenting certain integral representations for by choosing to use three known integral representations for the Riemann zeta function . Our method used here is similar to those in some earlier works, but our results seem a little simpler. Some relevant connections of some special cases of our results presented here with those in earlier works are also pointed out.
To do this, we first observe a simple property asserted in the following lemma.
Lemma 1 If some representations of the Riemann zeta function are analytic in a deleted neighborhood of , except for a simple pole at with its residue 1, then the following function defined by
is analytic at if we define
Furthermore, we have
Proof We prove only (1.20). If the above-defined is analytic at , then the Taylor series expansion of is given as follows:
in a neighborhood of . In view of (1.3), by uniqueness of Taylor (or Laurent) series expansion of a function, (1.20) is proved. The other argument is obvious from a well-known property of the Riemann zeta function . □
A well-known (and potentially useful) relationship between the polygamma functions and the generalized zeta function is also given by
In particular, we have
2 Integral representations for
We begin by presenting an integral representation for the Stieltjes constants given in the following theorem.
Theorem 1 The following integral representation for holds true:
We note that in (2.7) below is analytic in a neighborhood of . So we can use the relation (1.20) for the integral representation of . In this regard, we first try to get the following formulas asserted by Lemma 2 below.
Lemma 2 Each of the following formulas holds true:
where denotes the greatest integer less than or equal to a real number x.
Proof The formula (2.2) is obvious. For (2.3), we recall the Maclaurin series expansion of sint:
By using (2.4), we have
Similarly, we obtain
Now it is not difficult to combine the two formulas (2.5) and (2.6) to see the unified formula (2.3). □
To get the n th derivative of a product of the two involved functions in (2.7),
we apply Leibniz’s generalization of the product rule for differentiation and use the results in Lemma 2, in view of (1.20), to yield (2.1). □
In order to use (2.8) to get an integral representation for , we first find the following formula given in Lemma 3.
Lemma 3 If we define by
then we have a recurrence formula for
and an empty sum (as usual) is understood to be nil throughout this paper.
In addition to the formulas in (2.11), the next several are given as follows:
Proof of Lemma 3 Taking the logarithmic derivative of , we have
Using Leibniz’s generalization of the product rule for differentiation when we differentiate the last formula k times and taking the limit on the resulting identity, and applying (1.22) to the last resulting formula, we obtain
This completes the proof of Lemma 3. □
Using Leibniz’s generalization of the product rule for differentiation to differentiate both sides of in (2.8) with respect to s, n times, and taking the limit , being analytic at on the resulting identity, and finally using the in (2.9) and the relation (1.20), we obtain an integral formula for asserted by Theorem 2 below.
Theorem 2 The following integral representation for holds true:
where are given in Lemma 3.
The first three of in (2.14) are given in Corollary 1 below.
Corollary 1 Each of the following integral formulas holds true:
Proof It is enough to apply (2.13) and a known recurrence formula (see, e.g., [, pp.369-371]) for
to the first three of in (2.14). For easy reference, we record here the first three of :
Applying Leiniz’s generalization of the product rule for differentiation to (2.20), similarly as in Theorems 1 and 2, we get an integral representation for given in Theorem 3 below.
Theorem 3 The following integral representation for holds true:
The first three of in (2.22) are given in Corollary 2 below.
Corollary 2 Each of the following integral formulas holds true:
Remark Setting in (2.1), in view of relation (1.21), we obtain an integral representation for γ:
which is a known formula (see, e.g., [, Eq. (3.67)]). Equation (1.7) is equal to Equation (2.21), which is recorded, for example, in [, p.17, Eq. (31)]. Equation (2.13) is a known result (see, e.g., [, p.355, Entry 3.427-2]). The result (2.24) is equal to the special case of (1.11) when . Connon’s result (1.17) is equal to the integral representation (2.15) for . It is also interesting to compare Connon’s result with our one (2.14).
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The author would like to express his deep gratitude for the reviewers’ helpful comments via their rather detailed reading to make this paper more clear. 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 (2010-0011005).
The author declares that he has no competing interests.
About this article
- gamma function
- Riemann zeta function
- Hurwitz (or generalized) zeta function
- psi (or digamma) function
- polygamma functions
- Euler-Mascheroni constant
- Stieltjes constants