Abstract
In this paper, we give asymptotic expansions and inequalities related to the generalized Somos quadratic recurrence constant, using its relation with the generalized Euler constant.
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1 Introduction
Somos’ quadratic recurrence constant is defined (see [1–3]) by
or
The constant σ arises in the study of the asymptotic behavior of the sequence
with the first few terms
This sequence behaves as follows (see [4, p. 446] and [3, 5]):
The constant σ appears in important problems from pure and applied analysis, and it is the motivation for a large number of research papers (see, for example, [1, 6–16]).
Sondow and Hadjicostas [15] introduced and studied the generalized-Euler-constant function \(\gamma (z)\), defined by
where the series converges when \(\vert z \vert \leq 1\). Pilehrood and Pilehrood [13] considered the function \(z\gamma (z)\) (\(\vert z \vert \leq 1\)). The function \(\gamma (z)\) generalizes both Euler’s constant \(\gamma (1)\) and the alternating Euler constant \(\ln \frac{4}{\pi } = \gamma (-1)\) [17, 18].
Sondow and Hadjicostas [15] defined the generalized Somos constant
Coffey [19] gave the integral and series representations for \(\ln \sigma_{t}\):
and
in terms of the polylogarithm function.
It is known (see [15]) that
Thus, formula (1.5) is closely related to Somos’ quadratic recurrence constant σ.
Define
Mortici [11] proved that for \(n\in \mathbb{N}\),
and
Lu and Song [10] improved Mortici’s results and obtained the inequalities:
and
for \(n\in \mathbb{N}\).
You and Chen [16] further improved inequalities (1.10)–(1.13). Recently, Chen and Han [7] gave new bounds for \(\gamma (1/2)-\gamma_{n}(1/2)\):
for \(n\in \mathbb{N}\), and presented the following asymptotic expansion:
as \(n\to \infty \). Moreover, these authors gave a formula for successively determining the coefficients in (1.15).
Chen and Han [7] pointed out that the lower bound in (1.14) is for \(n\geq 24\) sharper than the one in (1.12), and the upper bound in (1.14) is for \(n\geq 18\) sharper than the one in (1.12),
For any positive integer \(m\geq 2\), in this paper we give the asymptotic expansion of \(\gamma ( 1/m ) -\gamma_{n} ( 1/m ) \) as \(n\to \infty \). Based on the result obtained, we establish the inequality for \(\gamma ( 1/4 ) -\gamma_{n} ( 1/4 ) \). We also consider the asymptotic expansion for \(\gamma ( -1 ) - \gamma_{n} ( -1 ) \).
2 Lemmas
Lemma 2.1
As \(x\to \infty \),
where \(A(x)\) is defined by
with the coefficients \(a_{j}\) given by the recurrence relation
Here, and throughout this paper, an empty sum is understood to be zero.
Proof
Using the Maclaurin series of \(\ln (1+t)\),
we obtain
In view of (2.4), we can let
where \(a_{j}\) are real numbers to be determined.
Write (2.5) as
Direct computation yields
It follows from (2.4), (2.6), and (2.7) that
Equating coefficients of the term \(x^{-j}\) on both sides of (2.8) yields
For \(j=m\), we obtain \(a_{m}=\frac{(-1)^{m}}{m-1}\), and for \(j\geq m+1\), we have
We then obtain the recursive formula
which can be written as (2.3). The proof of Lemma 2.1 is complete. □
Lemma 2.2
Let
Then, for \(x\geq 1\),
Proof
It is well known that for \(-1< t\leq 1\) and \(m\in \mathbb{N}\),
which implies that for \(x\geq 1\) and \(m\geq 2\),
Using (2.12), we find that
and
The proof of Lemma 2.2 is complete. □
Remark 2.1
Using the methods from [20–22] it is possible to get estimations (based on the power series expansions) of the logarithm function that can be used, for example, in the analysis of parameterized Euler-constant function, which will be an item for further work.
Lemma 2.3
As \(x\to \infty \), we have
where \(C(x)\) is defined by
with the coefficients \(c_{j}\) given by the recurrence relation
Proof
In view of (2.4), we can let
where \(c_{j}\) are real numbers to be determined. Write (2.16) as
Noting that (2.7) holds, we have
Equating coefficients of the term \(x^{-j}\) on both sides of (2.17) yields
For \(j=2\), we obtain \(c_{2}=1/4\), and for \(j\geq 3\) we have
We then obtain the recursive formula (2.15). The proof of Lemma 2.3 is complete. □
The first few coefficients \(c_{j}\) are
3 Main results
For any positive integer \(m\geq 2\), Theorem 3.1 gives the asymptotic expansion of \(\gamma ( 1/m ) -\gamma_{n} ( 1/m ) \) as \(n\to \infty \).
Theorem 3.1
For any positive integer \(m\geq 2\), we have
where \(A(x)\) is given in (2.2). Namely,
Proof
Write (2.1) as
where
with the coefficients \(a_{j}\) given by the recurrence relation (2.3). From (3.3), we have
Adding (3.5) from \(k=n+1\) to \(k=\infty \), we have
which can be written as (3.1). The proof of Theorem 3.1 is complete. □
Remark 3.1
For \(m=2\) in (3.2), we obtain (1.15). For \(m=3\) in (3.2), we find
For \(m=4\) in (3.2), we find
Formula (3.7) motivated us to establish Theorem 3.2.
Theorem 3.2
For \(n\in \mathbb{N}\),
Proof
From the double inequality (2.11), we have
where \(a(x)\) and \(b(x)\) are given in (2.10). Adding inequalities (3.9) from \(k=n+1\) to \(k=\infty \), we have
which can be written as (3.8). The proof of Theorem 3.2 is complete. □
Remark 3.2
Inequality (3.8) can be further refined by inserting additional terms on both sides of the inequality. For example, for \(n\in \mathbb{N}\), we have
Remark 3.3
Following the same method as the one used in the proof of Theorem 3.2, we can prove the following inequality:
for \(n\in \mathbb{N}\). We omit the proof.
In view of (1.14), (3.11), (3.8), and (3.10), we pose the following conjecture.
Conjecture 3.1
For any positive integer \(m\geq 2\), we have
with the coefficients \(a_{j}\) given in (2.3).
By using the Maple software, we find, as \(n\to \infty \),
and
From a computational viewpoint, formulas (3.13), (3.14), and (3.15) improve formulas (1.15), (3.6), and (3.7), respectively.
For any positive integer \(m\geq 2\), we here provide a pair of recurrence relations for determining the constants \(p_{\ell }\equiv p_{\ell }(m)\) and \(q_{\ell }\equiv q_{\ell }(m)\) (see Remark 3.4) such that
as \(n\to \infty \). This develops formulas (3.13), (3.14), and (3.15) to produce a general result given by Theorem 3.3.
Theorem 3.3
For any positive integer \(m\geq 2\), we have
as \(n\to \infty \), where \(\lambda_{\ell }\equiv \lambda_{\ell }(m)\) and \(\mu_{\ell }\equiv \mu_{\ell }(m)\) are given by a pair of recurrence relations
and
with
Here \(a_{j}\) are given in (2.3).
Proof
In view of (3.13), (3.14), and (3.15), we let
where \(\lambda_{\ell }\) and \(\mu_{\ell }\) are real numbers to be determined. This can be written as
Direct computation yields
which can be written as
Substituting (3.21) into (3.20) we have
On the other hand, it follows from (3.1) that
Equating coefficients of the term \(n^{-j}\) on the right-hand sides of (3.22) and (3.23), we obtain
Setting \(j=2\ell -1\) and \(j=2\ell \) in (3.24), respectively, yields
and
For \(\ell =1\), from (3.25) and (3.26) we obtain
and for \(\ell \geq 2\) we have
and
We then obtain the recurrence relations (3.18) and (3.19). The proof of Theorem 3.3 is complete. □
Here we give explicit numerical values of some first terms of \(\lambda_{\ell }\) and \(\mu_{\ell }\) by using formulas (3.18) and (3.19). This shows how easily we can determine the constants \(a_{\ell }\) and \(b_{\ell }\) in (3.17).
Remark 3.4
The constants \(p_{\ell }\) and \(q_{\ell }\) in (3.16) are given by
Setting \(m=2, 3\), and 4 in (3.16), respectively, yields (3.13), (3.14), and (3.15).
Noting that \(\ln \frac{4}{\pi } = \gamma (-1)\) holds, Theorem 3.4 presents the asymptotic expansion for \(\ln \frac{4}{\pi }\).
Theorem 3.4
As \(n\to \infty \), we have
where \(C(x)\) is given in (2.14). Namely,
Proof
Write (2.13) as
where
with the coefficients \(c_{j}\) given by the recurrence relation (2.15).
From (3.29), we have
Adding (3.31) from \(k=n+1\) to \(k=\infty \), we have
which can be written as (3.27). The proof of Theorem 3.4 is complete. □
Remark 3.5
We see from (3.28) that the alternating Euler constant \(\ln \frac{4}{\pi }\) has the following expansion:
4 Conclusions
In this paper, we give asymptotic expansions related to the generalized Somos quadratic recurrence constant (Theorems 3.1 and 3.3). We present the inequalities for \(\gamma ( \frac{1}{4} ) - \gamma_{n} ( \frac{1}{4} ) \) and \(\gamma ( \frac{1}{3} ) - \gamma_{n} ( \frac{1}{3} ) \) (see (3.8), (3.10), and (3.11)). The expansion of the alternating Euler constant \(\ln \frac{4}{\pi }\) is also obtained (see (3.33)).
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Ma, XS., Chen, CP. Inequalities and asymptotic expansions related to the generalized Somos quadratic recurrence constant. J Inequal Appl 2018, 147 (2018). https://doi.org/10.1186/s13660-018-1741-8
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DOI: https://doi.org/10.1186/s13660-018-1741-8