1. Introduction

Let It is well known that the sequence converges to Euler's constant , where Nothing is known on the algebraic background of such mathematical constants like Euler's constant . So we are interested in better diophantine approximations of these numbers, particularly in rational approximations.

In 1995 the author [1] introduced a linear transformation for the series with integer coefficients which improves the rate of convergence. Let be an additional positive integer parameter.

Proposition 1.1 (see [1]). For any integers and one has Particularly, by choosing , one gets the following result.

Corollary 1.2. For any integer one has

Some authors have generalized the result of Proposition 1.1 under various aspects. At first one cites a result due to Rivoal [2].

Proposition (see [2]). For tending to infinity, one has Kh. Hessami Pilehrood and T. Hessami Pilehrood have found some approximation formulas for the logarithms of some infinite products including Euler's constant . These results are obtained by using Euler-type integrals, hypergeometric series, and the Laplace method [3].

Proposition ([3]). For tending to infinity the following asymptotic formula holds:

Recently the author has found series transformations involving three parameters , and , [4]. In Propositions 1.5 and 1.6 certain integral representations of the (discrete) series transformations are given, which exhibit important (analytical) tools to estimate the error terms of the transformations.

Proposition 1.5 (see [4]). Let , and be integers. Additionally one assumes that Then one has

Proposition (see [4]). Let , and be integers. Additionally one assumes that Then one has with

Setting one gets an explicit upper bound from Proposition 1.6

Corollary 1.7. For integers , , one has where is some constant depending only on . For one gets For an application of Corollary 1.7 let the integers and be defined by denotes the von Mangoldt function. By [5, Theorem  434] one has Then, for , there is some integer such that Multiplying (1.14) by , we deduce the following corollary.

Corollary. There is an integer such that one has for all integers that

2. Results on Rational Approximations to

In 2007, Aptekarev and his collaborators [6] found rational approximations to , which are based on a linear third-order recurrence. For the sake of brevity, let .

Proposition (see [6]). Let and be two solutions of the linear recurrence with , , and , , . Then, one has , , and with two positive constants . It seems interesting to replace the fraction by and to estimate the remainder in terms of .

Corollary 2.2. Let . Then there are two positive constants , such that for all sufficiently large integers one has

Recently, Rivoal [7] presented a related approach to the theory of rational approximations to Euler's constant , and, more generally, to rational approximations for values of derivatives of the Gamma function. He studied simultaneous Padé approximants to Euler's functions, from which he constructed a third-order recurrence formula that can be applied to construct a sequence in that converges subexponentially to for any complex number . Here, is defined by its principal branch. We cite a corollary from [7].

Proposition 2.3 (see [7]). (i) The recurrence provides two sequences of rational numbers and with , , and , , such that converges to .(ii) The recurrence provides two sequences of rational numbers and with , , and , , such that converges to .

The goal of this paper is to construct rational approximations to without using recurrences by a new application of series transformations. The transformed sequences of rationals are constructed as simple as possible, only with few concessions to the rate of convergence (see Theorems 2.4 and 6.2 below).

In the following we denote by the Bernoulli numbers, that is, , , , and so on (In Sections 3–6 the Bernoulli numbers cannot be confused with the integers from Corollary 2.2.) In this paper we will prove the following result.

Theorem 2.4. Let , , and be positive integers, and Then, where is some positive constant depending only on .

3. Proof of Theorem 2.4

Lemma 3.1. One has for positive integers and

Proof. Applying the well known inequality , we get This proves the lemma.

takes its maximum value for with which leads to a better bound than in Lemma 3.1. But we are satisfied with Lemma 3.1. A main tool in proving Theorem 2.4 is Euler's summation formula in the form where is a suitable chosen parameter, and the remainder is defined by a periodic Bernoulli polynomial , namely with Applying the summation formula to the function , we get (see [8, equation ( 5)] ) It follows that We prove Theorem 2.4 for . The case is treated similarly. So we have again by the above summation formula that First, we estimate the integral on the right-hand side of (3.8). We have since . Next, we assume that . Hence , and therefore we estimate the integral on the right-hand side in (3.9) by In the sequel we put . Moreover, in the above formula we now replace by with . In order to estimate we use Stirling's formula Then, it follows that and similarly we have By using the definition of in Theorem 2.4, the formula (1.1) for , and the identities (3.8), (3.9), it follows that where is specified to and to . Moreover, we know from [4, Lemma  2] that By setting , the above formula for the series transformation of simplifies to where , and . Here, we have used the results from Corollary 1.7, (3.13), and (3.14). The sum vanishes, since for every real number we have where on the right-hand side for an integer with one term in the numerator equals to zero.

The inequality holds for all integers . Now, using Lemma 3.1, we estimate the right-hand side in (3.17) for and as follows: The last but one estimate holds for all integers , , and is a suitable positive real constant depending on . This completes the proof of Theorem 2.4.

4. On the Denominators of

In this section we will investigate the size of the denominators of our series transformations