Certain integral representations of Stieltjes constants
© Choi; licensee Springer. 2013
Received: 20 August 2013
Accepted: 22 October 2013
Published: 12 November 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
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.
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 . □
2 Integral representations for
We begin by presenting an integral representation for the Stieltjes constants given in the following theorem.
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.
where denotes the greatest integer less than or equal to a real number x.
Now it is not difficult to combine the two formulas (2.5) and (2.6) to see the unified formula (2.3). □
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.
and an empty sum (as usual) is understood to be nil throughout this paper.
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.
where are given in Lemma 3.
The first three of in (2.14) are given in Corollary 1 below.
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.
The first three of in (2.22) are given in Corollary 2 below.
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).
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).
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