Relations among Sums of Reciprocal Powers — Part II

José Marı́a Amigó Centro de Investigación Operativa, Universidad Miguel Hernández, Avenida de la Universidad s/n, 03202 Elche (Alicante), Spain Correspondence should be addressed to José Marı́a Amigó, jm.amigo@umh.es Received 18 July 2008; Accepted 10 September 2008 Recommended by Pentti Haukkanen Some formulas relating the classical sums of reciprocal powers are derived in a compact way by using generating functions. These relations can be conveniently written by means of certain numbers which satisfy simple summation formulas. The properties of the generating functions can be further used to easily calculate several series involving the classical sums of reciprocal powers. Copyright q 2008 José Marı́a Amigó. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Introduction
In 1 , we studied some arithmetic relations among the classical numbers: In this paper, we extend this analysis to the remaining Although the numbers λ n , ζ n , and η n are related to each other through the identities η 1 ln 2 and thus we will use all of them in order to keep the algebraic expressions as simple as possible.

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For k ≥ 2, define It will be shown below that these numbers can be alternatively expressed as if n 0, the first term on the right hand side of 1.7 has to be dropped .This and other formulas expressing the numbers D k by means of sums of reciprocal powers with odd arguments and, conversely, η 2n 1 , ζ 2n 1 , and λ 2n 1 via D k will be proved in Section 2 see Propositions 2.2 and 2.4, and Corollary 2.6 below using some generating functions defined by the numbers λ n , η n , ζ n , and D n .The properties of these generating functions, which are given as expansions both in powers and in partial fractions, will be instrumental for most of the subsequent results.
Sections 3, 4 deal with the numbers D k and with the classical sums of reciprocal powers, respectively.In particular, Section 3 is mainly devoted to the calculation of several series containing D k .In Section 4, the focus changes onto some particular series whose terms contain λ-, η-, ζ-, and L-numbers like, for example, Finally, in the brief Section 5, the generating function of the numbers D k is expressed using the Psi Digamma function ψ x Γ x /Γ x .

Main statements
Define the generating functions Λ x , E x , and Z x by where, in principle, x ∈ C since lim n→∞ λ n lim n→∞ η n lim n→∞ ζ n 1, these formal power series converge only for |x| < 1 .Furthermore, denote by Λ x and Λ − x the even and odd parts, respectively, of Λ x and similarly for E ± x and Z ± x .Then, the following identities hold 3, 4.3.67/68/70:

2.4
Owing to 1.3 and 1.4 , the above generating functions fulfill the trivial relations respectively.
On substituting the definitions of λ n , η n , and ζ n into the corresponding generating functions, we find the following expansions in partial fractions:

2.6
In particular, the expansions will be used below.
where according to the Taylor expansion 2.2 ,

2.12
Proof.For each ν ∈ N, we have upon k − 1 integrations by parts

2.13
Sum now both sides on ν, 1 ≤ ν ≤ N, and use the identity 4, 1.342 1 to get on the left side

2.15
To finish the proof, let N → ∞ and use the Riemann-Lebesgue lemma.
In particular, Define next the generating function

2.19
Owing to the vanishing rate of the coefficients D n , this power series is convergent for all x ∈ C.Then,

2.20
Furthermore, let D x and D − x be the even and odd parts of D x , that is,

2.21
Thus, if x is meant to be real, D x and 1/i D − x are the real and imaginary part, respectively, of D x .

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Proposition 2.3.The following relations hold:

2.22
Proof.We claim that

2.23
In fact, from Comparison with 2.7 proves the claim.Take now even and odd parts.
Solving for E x and Z x in 2.22 , we get the identities which lead to a kind of converse of Proposition 2.2.

2.29
2 The second identity follows from 2.27 since use ζ 0

2.30
Remark 2.5.Equation 1.7 is nothing else but the formula

Summation formulas for the D k s
Before deriving more relations involving the sums of reciprocal powers, we obtain next some "summation" formulas for the numbers D k .Two integral summation formulas follow trivially from the very definition of the generating function D x and 2.20 , namely, Other similar series can be also straightforwardly deduced after differentiating 2.21 , and substituting fixed values for x.In particular, the series where cos 2t 1 − 2 sin 2 t, 1.6 and 2.16 were used will be needed below.Note that from 3.7 and 3.4 , it follows Furthermore, from 2.20 , we have 3.9 so, after separating real and imaginary parts, the equations hold for x ∈ R. Letting x → 1/2 one recovers 3.2 and 3.4 .

3.15
Comparison with 3.10 , that is,

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2 Analogously to 1 , the summation formula follows comparing 3.17 with 3.11 , that is, where |x| < 1.
Of course, all these summation formulas can be also checked using the integral representation 1.6 .Finally, a finite summation formula can be derived in the following way.

3.19
Therefore, changing the variable in the integral, we get 3.20

Further relations
From the close-form expressions obtained in the previous sections, we can derive a number of interesting results for series containing, in turn, the series ζ n , λ n , or η n .For completeness, we will also include the series L n , what requires the consideration of a new generating function C x defined in 1 .As a first example, we will prove the following proposition.

4.1
Note, in particular, the series Proof. 1 Setting x 1/2 in 2.27 , we get Thus, by 3.3 , On the other hand see 2.4 , Adding up the even and odd parts, we get Hence, by 3.5 ,

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On the other hand, by 2.3 , Adding up the even and odd parts, we get

4.15
Adding up the even and odd parts,

4.19
Adding up the even and odd parts,

4.20
The next theorem shows that more challenging results can be achieved by being slightly more sophisticated.

4.21
where all the series start with n 1.

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Proof.We will proceed left to right and top to bottom. 1 From 2.4 and 2.27 , it follows for |x| < 1.Thus, by the Tauber theorem 6 , 3.2 and 3.7 ,

4.23
where we have used the L'Hopital rule in the last line.We will use L'Hopital's rule also in the sequel to resolve indeterminacies. 2 From 2.3 and 2.26 , it follows for 0 < |x| < 1.Thus, by Tauber's theorem, 3.4 and 3.7 ,

16
International Journal of Mathematics and Mathematical Sciences From 2.27 , 4.17 , and 3.7 , we get

4.38
International Journal of Mathematics and Mathematical Sciences 14 From 2.4 , 4.18 , and 4.28 , we get 4.44

Enter ψ x
Let ψ x , x ∈ C, be the logarithmic derivative of the Gamma function.Then, if |x| < 1 7, 1.17 5

Proposition 2 . 4 .
The numbers η 2n 1 and ζ 2n 1 , n ≥ 1, can be expressed in terms of the D k and the elementary values η 2k and ζ 2k by

−ψ 1 −
x − γ −ψ x − π cot πx − γ, ψ x ≡ ψ x ψ −x /2.International Journal of Mathematics and Mathematical SciencesSubstitution of these expressions for E x and Z x in 2.22 and comparison with 2 dt. 1.6 Indefinite integrals of this type were considered by Ramanujan 2, page 260 .The constants D k have the property of relating the values of ζ-numbers eventually, ηor λ-numbers with odd argument to the elementary values ζ 2n 2π 2n |B 2n |/2 2n !where B 2n are the Bernoullian numbers as, for example, in Equation 1.7 and also the first formula of Proposition 2.4 show that η 2n 1 and, for that case, ζ 2n 1 and λ 2n 1 can be expressed in terms of only D 2 , D 4 , . . ., D 2n 2 .Furthermore, adding the two formulas of Proposition 2.4 and using 1.4 , we obtain the following result.
1 x 2n , 2.32 and cot πx D − x was calculated in the last proof.For n 1 one gets the representation