Abstract
In this paper, we establish an asymptotic expansion for the Euler–Mascheroni constant. Based on this expansion, we establish a two-sided inequality and a continued fraction approximation for the Euler–Mascheroni constant.
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1 Introduction
Throughout this paper, \({\mathbb {N}}\) represents the set of positive integers and \({\mathbb {N}}_0:={\mathbb {N}}\cup \{0\}\). The Euler–Mascheroni constant \(\gamma =0.577215664\ldots \) is defined as the limit of the sequence
The Euler–Mascheroni constant is a number that appears in analysis and number theory. It is not known yet whether the number is irrational or transcendental [20, 25]. The following two-sided inequality was presented in [24, 28]:
which shows that the convergence of the sequence \(D_{n}\) to \(\gamma \) is very slow (like \(n^{-1}\)). By changing the logarithmic term in (1.1), faster approximation formulas to the Euler–Mascheroni constant were presented in [7, 10, 16, 17, 23]. For example, Chen and Mortici [7] established the following approximation formula:
and posed an open question as follows: For a given \(p\in {\mathbb {N}}_0\), find the constants \(a_{i}\) \((i=0, 1, 2,\ldots , p)\), such that
is the fastest sequence which would converge to \(\gamma \). This open problem has been considered by Yang [27], Gavrea and Ivan [18], and Lin [21]. Recently, Chen [5] determined the coefficients \(a_j\) and \(b_j\), such that
where \(p\in {\mathbb {N}}\), \(q\in {\mathbb {N}}_0\) and \(p=q+1\). This solves an open problem of Mortici [22]. There are a lot of formulas expressing \(\gamma \) as series, integrals, or products [4, 8, 9, 13, 14, 20]. For more information on the Euler–Mascheroni constant \(\gamma \), please refer to survey papers [15, 26] and expository book [19].
In this paper, we consider the sequence \((A_{n})_{n\in {\mathbb {N}}}\) defined by
Using the computer program MAPLE 13, we find, as \(n\rightarrow \infty \)
We here give a formula for determining these coefficients in the right-hand side of (1.5). Then, we establish a two-sided inequality and a continued-fraction approximation for the Euler–Mascheroni constant.
The numerical values given in this paper have been calculated via the computer program MAPLE 13.
2 Lemmas and Preliminaries
The gamma function may be defined by \( \Gamma (x)=\int _{0}^{\infty }t^{x-1}e^{-t}\textrm{d} t\,\, (x>0)\). The logarithmic derivative of the gamma function \(\psi (x)=\Gamma '(x)/\Gamma (x)\) is known as the psi (or digamma) function. The derivatives of the psi function \(\psi ^{(n)}(x)\,\, (n \in {\mathbb {N}})\) are called the polygamma functions. It is well known that the psi function has the duplication formula [1, p. 259, Eq. (6.3.8)]:
The following series expansion and asymptotic formula hold (see [1, p. 260]):
and
where \(B_n\) (\(n\in {\mathbb {N}}_0\)) are the Bernoulli numbers defined by
It follows from the known result (see [6, Eq. (3.26)]) that:
Lemma 1.1
(see [3, Theorem 9]) Let \(k \ge 1\) and \(n \ge 0\) be integers. Then, for all real numbers \(x>0\)
where
\(B_n\) are the Bernoulli numbers.
It follows from (2.5) that for \(x>0\):
Using the recurrence formula
we deduce from (2.6) that for \(x>0\)
Lemma 1.2
(see [2]) For \(x>\frac{1}{2}\) and \(N\in {\mathbb {N}}_0\)
where \(B_n(t)\) are the Bernoulli polynomials defined by the following generating function:
Noting that
we obtain from (2.9) that for \(x>0\)
Lemma 1.3
(see [11, 12]) Let \(a_1\not =0\) and \( A(x)\sim \sum _{j=1}^\infty \frac{a_j}{x^{j}}\, (x\rightarrow \infty )\) be a given asymptotic expansion. Then, the function \(B(x):=a_1/A(x)\) has asymptotic expansion of the following form:
where
Remark 2.4
Lemma 2.3 provides a method to construct a continued-fraction approximation based on a given asymptotic expansion. The details of this method are given below.
Let \(a_1\not =0\) and
be a given asymptotic expansion. Then, the asymptotic expansion (2.11) can be transformed into the continued-fraction approximation of the form
wherein the constants are given by the following recurrence relations:
Clearly, since \(a_j \Longrightarrow b_j \Longrightarrow c_j \Longrightarrow d_j \Longrightarrow \cdots \), the asymptotic expansion (2.11) is transformed into the continued-fraction approximation (2.12), and the constants in the right-hand side of (2.12) are determined by the system (2.13).
3 Asymptotic Expansion
We provide a recurrence relation for successively determining the coefficients of \(\frac{1}{n^k}\) \((k\in {\mathbb {N}})\) in expansion (1.5) given by Theorem 3.1.
Theorem 1.1
The sequence \((A_{n})_{n\in {\mathbb {N}}}\), defined by (1.4), has the asymptotic expansion
with the coefficients \(a_k\) given by the recurrence relation
where \(B_n\) are the Bernoulli numbers.
Proof
Denote
In view of (1.5), we can let \(I_n\sim J_n\) and
as \(n\rightarrow \infty \), where \(a_k\) are real numbers to be determined.
We obtain by (2.2) that
Using (3.3) and (2.7), we have
We obtain from (2.1) that
We obtain by (3.5) and (2.7) that
Substituting (3.6) into (3.4), we obtain
Using (2.3) and (2.4), we find as \(n\rightarrow \infty \)
which can be written as
Direct computation yields
We then obtain
Equating coefficients of the term \(n^{-k}\) on the right-hand sides of (3.8) and (3.9) yields
for \(k\ge 2\). For \(k=2\) we obtain \(a_1=\frac{1}{2}\), and for \(k\ge 3\), we have
which gives the desired result (3.2). The proof of Theorem 3.1 is complete. \(\square \)
4 A Two-Sided Inequality
Motivated by (1.5), we establish a two-sided inequality for the Euler–Mascheroni constant given by Theorem 4.1.
Theorem 1.2
Let the sequence \((A_{n})_{n\in {\mathbb {N}}}\) be defined by (1.4). Then, for \(n\ge 1\)
Proof
We consider the sequences \(\left( x_{n}\right) _{n\in {\mathbb {N}}}\) and \(\left( y_{n}\right) _{n\in {\mathbb {N}}}\) defined by
Clearly
Noting that (3.7), we obtain by (2.8) and (2.10) that
and
where
and
We then obtain \(x_{n+1}<x_{n}\) and \(y_{n+1}>y_{n}\) for \(n\ge 2\). Direct computations give
We see that the sequence \((x_{n})\) is strictly decreasing and \((y_{n})\) is strictly increasing for \(n \ge 1\), and we have
and
The proof of Theorem 4.1 is complete. \(\square \)
5 Continued-Fraction Approximation
We convert the asymptotic expansion (3.1) into a continued-fraction approximation given by Theorem 5.1.
Theorem 1.3
It is asserted that
Proof
By Remark 2.4, we can convert (3.1) into a continued-fraction approximation of the form
where the constants in the right-hand side can be determined by using the system (2.13). We see from (1.5) that
We obtain from the first recurrence relation in (2.13) that
We obtain from the second recurrence relation in (2.13) that
Continuing the above process, we get
The proof of Theorem 5.1 is thus completed. \(\square \)
Remark 5.2
Based on (5.1), we find the following two-sided inequality:
Following the same method as was used in the proof of Theorem 4.1, we can prove (5.2). We here omit the proof. Elementary calculations show that
and
Hence, for \(n\ge 2\), the two-sided inequality (5.2) is more accurate than the two-sided inequality (4.1).
References
Abramowitz, M., Stegun, I.A. (eds.): Handbook of Mathematical Functions with Formulas, Graphs\(,\) and Mathematical Tables, Applied Mathematics Series 55. Ninth Printing, National Bureau of Standards, Washington, D.C. (1972)
Allasia, G., Giordano, C., Pećarić, J.: Inequalities for the gamma function relating to asymptotic expansions. Math. Inequal. Appl. 5(3), 543–555 (2002)
Alzer, H.: On some inequalities for the gamma and psi functions. Math. Comput. 66, 373–389 (1997)
Chen, C.-P.: Inequalities for the Lugo and Euler-Mascheroni constants. Appl. Math. Lett. 25(4), 787–792 (2012)
Chen, C.-P.: Approximation formulas and inequalities for the Euler-Mascheroni constant, Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, 115(2), Article 56 (2021). https://doi.org/10.1007/s13398-021-00999-4
Chen, C.-P., Choi, J.: Inequalities and asymptotic expansions for the constants of Landau and Lebesgue. Appl. Math. Comput. 248, 610–624 (2014)
Chen, C.-P., Mortici, C.: New sequence converging towards the Euler-Mascheroni constant. Comput. Math. Appl. 64, 391–398 (2012)
Chen, C.-P., Srivastava, H.M.: New representations for the Lugo and Euler-Mascheroni constants. Appl. Math. Lett. 24(7), 1239–1244 (2011)
Chen, C.-P., Srivastava, H.M.: New representations for the Lugo and Euler-Mascheroni constants, II. Appl. Math. Lett. 25(3), 333–338 (2012)
Chen, C.-P., Srivastava, H.M., Li, L., Manyama, S.: Inequalities and monotonicity properties for the psi (or digamma) function and estimates for the Euler-Mascheroni constant. Integral Transforms Spec. Funct. 22, 681–693 (2011)
Chen, C.-P., Srivastava, H.M., Wang, Q.: A method to construct continued fraction approximations and its applications, Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, 115(3), Article 97 (2021)
Chen, C.-P., Wang, Q.: Asymptotic expansions and continued fraction approximations for the harmonic numbers. Appl. Anal. Discrete Math. 13(2), 569–582 (2019)
Choi, J.: Some mathematical constants. Appl. Math. Comput. 187, 122–140 (2007)
Choi, J., Srivastava, H.M.: Integral representations for the Euler-Mascheroni constant \(\gamma \). Integral Transforms Spec. Funct. 21(9), 675–690 (2010)
Dence, T.P., Dence, J.B.: A survey of Euler’s constant. Math. Mag. 82, 255–265 (2009)
DeTemple, D.W.: The non-integer property of sums of reciprocals of consecutive integers. Math. Gaz. 75, 193–194 (1991)
DeTemple, D.W.: A quicker convergence to Euler’s constant. Am. Math. Mon. 100, 468–470 (1993)
Gavrea, I., Ivan, M.: Optimal rate of convergence for sequences of a prescribed form. J. Math. Anal. Appl. 402, 35–43 (2013)
Havil, J.: Gamma: exploring Euler’s constant. Princeton University Press, Princeton (2003)
Lagarias, J.C.: Euler’s constant: Euler’s work and modern developments. Bull. Am. Math. Soc. 50(4), 527–628 (2013)
Lin, L.: Asymptotic formulas associated with psi function with applications. J. Math. Anal. Appl. 405, 52–56 (2013)
Mortici, C.: On new sequences converging towards the Euler-Mascheroni constant. Comput. Math. Appl. 59, 2610–2614 (2010)
Negoi, T.: A faster convergence to the constant of Euler. Gazeta Matematică, seria A 15, 111–113 (1997). ((in Romanian))
Rippon, P.J.: Convergence with pictures. Am. Math. Mon. 93, 476–478 (1986)
Sondow, J.: Criteria for irrationality of Euler’s constant. Proc. Am. Math. Soc. 131(11), 3335–3345 (2003)
Srivastava, H.M.: A survey of some recent developments on higher transcendental functions of analytic number theory and applied mathematics. Symmetry 13 2294, 1–22 (2021)
Yang, S.: On an open problem of Chen and Mortici concerning the Euler-Mascheroni constant. J. Math. Anal. Appl. 396, 689–693 (2012)
Young, R.M.: Euler’s constant. Math. Gaz. 75, 187–190 (1991)
Acknowledgements
This work was the Fundamental Research Funds for the Universities of Henan Province (NSFRF210446).
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Han, XF., Chen, CP. Approximations to the Euler–Mascheroni Constant. Bull. Iran. Math. Soc. 49, 76 (2023). https://doi.org/10.1007/s41980-023-00820-5
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DOI: https://doi.org/10.1007/s41980-023-00820-5