Abstract

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.

1. Introduction

In [1], we studied some arithmetic relations among the classical numbers:In this paper, we extend this analysis to the remainingAlthough the numbers and are related to each other through the identities and thuswe will use all of them in order to keep the algebraic expressions as simple as possible.

For , defineIt will be shown below that these numbers can be alternatively expressed asIndefinite integrals of this type were considered by Ramanujan [2, page 260]. The constants have the property of relating the values of -numbers (eventually, - or -numbers) with odd argument to the elementary values (where are the Bernoullian numbers) as, for example, in (if the first term on the right hand side of (1.7) has to be dropped). This and other formulas expressing the numbers by means of sums of reciprocal powers with odd arguments and, conversely, , and via 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 , , , and . 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 and with the classical sums of reciprocal powers, respectively. In particular, Section 3 is mainly devoted to the calculation of several series containing . In Section 4, the focus changes onto some particular series whose terms contain , and -numbers like, for example,

Finally, in the brief Section 5, the generating function of the numbers is expressed using the Psi (Digamma) function

2. Main Statements

Define the generating functions and bywhere, in principle, (since these formal power series converge only for ). Furthermore, denote by and the even and odd parts, respectively, of and similarly for and Then, the following identities hold [3, 4.3.67/68/70]:

Owing to (1.3) and (1.4), the above generating functions fulfill the trivial relationsrespectively.

On substituting the definitions of , and into the corresponding generating functions, we find the following expansions in partial fractions:In particular, the expansionswill be used below.

Proposition 2.1. The integral representation (1.6) holds.

Proof. Let . By repeated partial integration, one getswhere according to the Taylor expansion (2.2),Hence,

Since, for and , it follows that faster (in fact, much faster) than .

Proposition 2.2. The numbers can be evaluated in terms of , supplemented with if is odd, by the formula

Proof. For each , we have upon integrations by partsSum now both sides on and use the identity [4, 1.342(1)]to get on the left side
To finish the proof, let and use the Riemann-Lebesgue lemma.

In particular,as in [4, 3.747(7)], andas in [5, 4.2(3)]. Other expressions for different from (2.12) can be found in [4, 3.748(2)].

Due to the definition (1.5) and formula (2.12), each number embodies a relation among all elementary values and a finite number of their nonelementary counterparts , namely,for

Define next the generating functionOwing to the vanishing rate of the coefficients this power series is convergent for all Then,Furthermore, let and be the even and odd parts of that is, Thus, if is meant to be real, and are the real and imaginary part, respectively, of

Proposition 2.3. The following relations hold

Proof. We claim thatIn fact, fromwe obtainComparison with (2.7) proves the claim. Take now even and odd parts.

Solving for and in (2.22), we get the identities which lead to a kind of converse of Proposition 2.2.

Proposition 2.4. The numbers and can be expressed in terms of the and the elementary values and by

Proof. (1) Equation (2.26) reads explicitly (use in (2.3)),
(2) The second identity follows from (2.27) since (use in (2.4)),

Remark 2.5. Equation (1.7) is nothing else but the formulawritten explicitly. Indeed,and was calculated in the last proof. For one gets the representation

Equation (1.7) and also the first formula of Proposition 2.4 show that (and, for that case, and ) can be expressed in terms of only Furthermore, adding the two formulas of Proposition 2.4 and using (1.4), we obtain the following result.

Corollary 2.6. For

For we recover (2.17).

3. Summation Formulas for the

Before deriving more relations involving the sums of reciprocal powers, we obtain next some “summation" formulas for the numbers . Two (integral) summation formulas follow trivially from the very definition of the generating function and (2.20), namely,Also, Other similar series can be also straightforwardly deduced after differentiating (2.21),and substituting fixed values for . In particular, the series(where , (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 haveso, after separating real and imaginary parts, the equations hold for . Letting one recovers (3.2) and (3.4).

Proposition 3.1. The following identities hold:
(1) For In particular, for (2) For

Proof. (1) In fact,Comparison with (3.10), that is,where yields the result.
(2) Analogously to (1), the summation formula follows comparingwith (3.11), that is,where

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. For Therefore, changing the variable in the integral, we get

4. 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 , or . For completeness, we will also include the series , what requires the consideration of a new generating function defined in [1]. As a first example, we will prove the following proposition.

Proposition 4.1. The following identities hold true:

Note, in particular, the series

Proof. (1) Setting in (2.27), we getThus, by (3.3),On the other hand (see (2.4)),Adding up the even and odd parts, we get
(2) Setting in (2.26), we getHence, by (3.5),On the other hand, by (2.3),Adding up the even and odd parts, we get
(3) In [1], it is proved that, for holds, whereHence,by (2.2), andsinceAdding up the even and odd parts,
(4) In [1], it is proved that for Hence,Adding up the even and odd parts,

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

Theorem 4.2. The following equalities hold true:where all the series start with .

Proof. We will proceed left to right and top to bottom.
(1) From (2.4) and (2.27), it followsfor Thus, by the Tauber theorem [6], (3.2) and (3.7),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 followsfor Thus, by Tauber's theorem, (3.4) and (3.7),
(3) Analogously, from (2.2) and (4.11),since (see (4.12))
(4) Using (4.18), (4.17), and (4.28), it follows
(5) From (2.27), (2.2), and (3.7), we get
(6) From (2.27), (2.3), and (3.7), we get
(7) From (2.27), (4.17), and (3.7), we get
(8) From (2.4), (4.11), and (4.28), we get
(9) From (4.11), (2.3), and (4.28), we get
(10) From (4.11), (4.17), and (4.28), we get
(11) From (2.4), (2.26), (3.4), and (3.7), we get
(12) From (2.2), (2.26), (3.4), and (3.7), we get
(13) From (4.17), (2.26), and (3.7), we get
(14) From (2.4), (4.18), and (4.28), we get
(15) From (2.27), (4.18), (3.7), (4.12), and (4.28), we get
(16) From (2.2), (4.18), and (4.28), we get
(17) From (4.11), (4.18), (4.27), and (4.12), we get
(18) From (4.18), (2.3), and (4.28), we get
(19) From (4.18), (2.26), (4.28), (3.4), and (3.7), we get

5. Enter

Let be the logarithmic derivative of the Gamma function. Then, if [7, 1.17(5)]hence [7, 1.7(11)]It followswhere

Substitution of these expressions for and in (2.22) and comparison with (2.21) yields respectively.

Acknowledgments

This work was completed during a research stay at the Centre de Recerca Matemàtica in Barcelona. The author would like to thank very much Professor Manuel Castellet for making this stay possible. The kind hospitality of the Centre is also acknowledged.