1102/25/2008
EVALUATION OF SOME SIMPLE EULER-TYPE SERIESKhristo N. Boyadzhiev
Department of Mathematics, Ohio Northern UniversityAda, Ohio 45810
ABSTRACT
We evaluate five series, three of which involve harmonic numbers and one involvesStirling numbers of the first kind. The evaluation of these series is reduced to the evaluation ofcertain integrals, including the moments of the polylogarithm. : 11M99, 33B30, 40G99
Keywords and phrases : Harmonic numbers, Stirling numbers, zeta function,polilogarithms, Euler’s beta function
1. INTRODUCTION
In this note we evaluate in closed form the Euler-type sums, , (1.1) , (1.2) , (1.3)where (1.4)are the generalized harmonic numbers with , and in (1.3) are integers. We also evaluate the series , (1.5)where is a fold convolution of harmonic numbers. The fifth series is, (1.6)where are the Stirling (cycle) numbers of the first kind as defined in [4]. This extends theinteresting representation of the Riemann zeta function . (1.7)The evaluation of the first three series is based on the evaluation of the integrals, (1.8)where is the polylogarithm (1.9) with when (see [7, p.106]).Using the identity (1.10)(see [4, 5.41], [6, p. 611] ), the series (1.1), (1.5) and (1.6) can be written in an obviousalternative form.The motivation for this work came from two problems by Ovidiu Furdiu - problem 854 inthe College J. Math., 38 (3), 2007, and problem H-653 in the Fibonacci Quarterly, 45 (1), 2007.
2. EVALUATION OF THE MOMENTS OF THE POLYLOGARITHM
The following is true.
Lemma 2.1.
For every positive integer p and every = (2.1) .Here is the Riemann zeta function, is the digamma function, and is Euler’sconstant. Proof . The first equality comes from term-wise integration using the series in (1.9). Then wewrite ,and thus is expressed in terms of . Repeating the procedure times bringsto (2.1) in view of the fact that , (2.2)(see [7, p. 14]). When is a positive integer, , (2.3)and therefore, . (2.4)In particular, for , . (2.5)(This computation is from [2]. The result also appeared in the recent online publication [3,4.4.100p] together with some other interesting integrals involving polylogarithms.)The integrals can be used to evaluate other integrals of similar type. For example,we have the immediate corollary.
Corollary 2.2.
For any positive integer , . (2.6)
3. EVALUATION OF THE FIRST THREE SERIES
The main tool in the evaluation of the series is Euler’s integral , (3.1)(see Euler’s Beta function in [6], [7]). We also need the generating function of the generalizedharmonic numbers , (3.2)( ). When we have and therefore, . (3.3)
Proposition 3.1.
For every two integers and , . (3.4)
For instance, when ,(For see (3.11) below).
Proof . Using (3.1), (3.2), and the fact that , (3.5)we compute , (3.6)and the proposition now follows from (2.6).Next we evaluate the series in (1.2).
Proposition 3.2.
For every (3.7)
Proof . We have according to (3.1), ,hence (3.8)and again we refer to (2.6) in order to complete the proof. Observation. Comparing the last integrals in (3.6) and (3.8) we notice that . (3.9)In the next section we describe a procedure for recursive evaluation of the sums for all. According to (3.9), this procedure applies also to the sums .
Proposition 3.3
For any three positive integers one has . (3.10)When , the sum on the right hand side is missing. Thus , (3.11)(3.12).
Proof.
Write first .Then .and the result follows from (2.1).
Remark 3.4.
The Euler sum (3.13)has been studied by many authors (starting with Euler himself) and its vales are known only for or odd (see [1]) . It is interesting that the similar sum (3.11), can be evaluated easilyfor every .
4. A RECURRENCE FOR THE SECOND SERIES (1.2)Lemma 4.1
For all we have the recursive relation, (4.1) whenever the three sums are convergent.
To see this, write and to obtain ,and the lemma follows immediately.We shall evaluate now for .Let first . From (3.1) we find . (4.2)Next, let . Then from (4.1) ,and since , (4.3)we have . (4.4)Now let . Then ,i.e. . (4.5)
5. CONVOLUTIONS OF HARMONIC NUMBERS AND STIRLING NUMBERS
In this section we evaluate the series defined in (1.5). For every two integers and set , (5.1)where are the harmonic numbers and . The convolutions are thecoefficients in the expansion (see (3.3)) . (5.2)Thus, when , . (5.3)
Proposition 5.1
For every and , . (5.4)
Also, . (5.5)
In particular, when we have the representation . (5.6)
Proof.
We multiply (3.1) by and sum for to obtain (according to (5.2))0 . (5.7)The last integral with the substitution turns into , (5.8)where (5.9)(for ) is the Hurwitz zeta function ([7]). Obviously, (5.10)which finishes the proof.Now we evaluate the last series from (1.6).
Proposition 5.2 . For every and , . (5.11)For the proof we use the generating function (see [4, p. 351]) . (5.12)Thus ,and we continue as in the proof of the previous proposition. The rest is left to the reader.
Remark 5.3 . The representation1(5.13)extend the remarkable formula , (5.14)as (5.13) turns into (5.14) when . Apparently, (5.14) was first obtained by Jordán Károly(Charles Jordan) [5, p.166, 195]. It has been rediscovered several times after that. Jordán Károly obtained also two similar extensions - see [5, p. 343] and also [7, p. 76] .Harmonic numbers are connected to Stirling numbers of the first kind as shown, forinstance, in [4] and [7, p.57]. This relationship is due to the similarity of their generating functions(3.3), (5.2) and (5.12). One such connection is listed below.
Corollary 5.4
From the above two propositions we conclude that for we have the identity , or . (5.15)
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