Approximations of the generalized-Euler-constant function and the generalized Somos’ quadratic recurrence constant

In this paper, we provide an estimate for approximating the generalized-Euler-constant function \(\gamma (z)=\sum_{k=1}^{\infty }z ^{k-1} (\frac{1}{k}-\ln \frac{k+1}{k} )\) by its partial sum \(\gamma _{N-1}(z)\) when \(0< z<1\). We obtain an asymptotic expansion for the generalized-Euler-constant function and show that the coefficients of the asymptotic expansion are explicitly expressed by the Eulerian fractions. Also, we find a recurrence relation for those coefficients. Using its relation with the generalized-Euler-constant function, we establish two inequalities for the generalized Somos’ quadratic recurrence constant. Moreover, two asymptotic expansions for the natural logarithm of the generalized Somos quadratic recurrence constant are presented.

Somos’ quadratic recurrence constant σ arises in the study of the asymptotic behavior of the sequence (see [8, p. 446] and [21, 27]):

where \(g_{n}\) are recursively defined by

with the first few terms

The constant σ is usually defined by

or

See [9, 20,21,22]. The constant σ appears in many important problems in pure and applied analysis and it has been investigated by a large number of researchers [3, 5, 9, 10, 12, 13, 15, 17,18,19, 25, 30].

Recently, Sondow and Hadjicostas [25] introduced and studied the generalized-Euler-constant function \(\gamma (z)\) defined by the power series

when \(|z|\leq 1\). There exist integral representations for the function

Its values include Euler’s constant \(\gamma =\gamma (1)\) and the “alternating Euler constant” \(\log \frac{4}{\pi }=\gamma (-1)\), see for example [23, 24]. In particular, at \(z=1/2\), the function takes the value

which is equivalent to

Mortici [15] proved that, for \(n\geq 1\), it follows that

where the partial sum of \(\gamma (z)\)

Lu and Song [13] improved Mortici’s estimate and proved that, for \(n\geq 1\),

You and Chen [30] improved these inequalities by using continued fraction. Very recently, Chen and Han [5] obtained new lower bounds for \(\gamma (1/2)-\gamma _{n}(1/2)\):

In their paper, Chen and Han pointed out that the lower bound in (1.11) is sharper than the one in (1.10) for \(n\geq 24\), and the upper bound in (1.11) is sharper than the one in (1.10) for \(n\geq 18\). Moreover, they gave the following asymptotic expansion:

with a recursive formula for successively determining the coefficients

Recently, there have been several interesting works related to approximations of Euler’s constant \(\gamma =\gamma (1)\), see for example [4, 16, 28] and the references therein. Also, some works related to approximations of \(\gamma (z)\) at special values \(1/2\), \(1/3\), and \(1/4\) appear in [3, 5, 13,14,15, 29, 30]. Motivated by these, the first aim of this paper is to give an approximation for \(\gamma (z)\) when \(0< z<1\). Specifically, we give a general inequality for the error bound of \(\gamma (z)-\gamma _{N-1}(z)\). The second aim is to establish an asymptotic expansion for \(\gamma (z)- \gamma _{N-1}(z)\) whose coefficients can be computed explicitly and recursively. In particular, we generalize inequality (1.11) and asymptotic expansion (1.12) due to Chen and Han. Using the relation between the generalized Somos’ quadratic recurrence constant and the generalized-Euler-constant function established by Sondow and Hadjicostas [25], we find approximate estimates for the generalized Somos’ quadratic recurrence constant and its natural logarithm, respectively. Moreover, two asymptotic expansions for the natural logarithm of the generalized Somos’ quadratic recurrence constant are presented.

Let us recall the following classical result [2, 7, 11]:

where \(A_{n}(z)\) is the Eulerian polynomial of degree n, and it can be defined by the exponential generating function

By (2.2), \(A_{n}(x)\) may be explicitly written as

where \(A(n,k)\) are known as the Eulerian numbers, the numbers of permutations of the numbers 1 to n in which exactly k elements are greater than the previous element, and they can be expressed by

Therefore, one may easily obtain

For \(z\neq 1\), denote the Eulerian fraction

From (2.2) it is clear that

which implies that \(b_{k}(z)\) can be computed by the following recurrence relation:

Also, we can obtain an explicit expression [11] as follows:

where \(S(k,j)\) are the Stirling numbers of the second kind.

It is worth noting that a general summation formula for power series using the Eulerian fractions was established in the recent paper [26].

Lemma 2.1

(cf. Theorem 4.2 [26])

Let m be a positive integer, and both P and Q be nonnegative integers. Suppose that \(f(x)\) has the \((m+1)\)th continuous derivative on \([P,Q]\). For \(0< z<1\), \(f^{(m)}(x)\) and \(f^{(m+1)}(x)\) keep opposite signs in \((P,Q)\). Then there exists a number \(\theta \in (0,1)\) such that

Remark 2.1

It is well known that the Euler–Maclaurin formula [1] is usually applied to treat the summation \(S=\sum_{k=a}^{b}f(k)\), and it was also used to approximate the generalized-Euler-constant function \(\gamma (z)\), see [12]. It seems that the above lemma is more suitable to treat a general summation formula for power series with the form \(S=\sum_{k=a}^{b}f(k)z^{k}\ (0< z<1)\). The lemma provides us a new approach to estimate \(\gamma (z)\), and it will play a key role in the next section.

Remark 2.2

In particular, if

as \(Q\rightarrow +\infty \), then (2.10) can be rewritten as

Obviously, taking \(f(x)=x^{n}\), \(P=0\), and \(m=n+1\), (2.12) reduces to (2.1).

Let N be a positive integer. In this section, by using (2.12) we first give a general inequality for the error bound of \(\gamma (z)-\gamma _{N-1}(z)\) when \(0< z<1\).

Theorem 3.1

Let p and q be any positive integers. If \(N\geq -\frac{c_{2p+3}(z)}{c _{2p+2}(z)}\), then we have

If \(N\geq -\frac{c_{2q+2}(z)}{c_{2q+1}(z)}\), then we have

In the above inequalities, the coefficients \(c_{k}(z)\) are explicitly expressed by

where \(b_{k}(z)\) are the Eulerian fractions introduced in the second section.

Proof

Write

After simple calculations, we arrive at

which implies that

By (3.4) and (3.6), for each \(i\geq 0\), we also have

Therefore, according to (2.12), for \(0< z<1\), there exists \(\theta \in (0,1)\) such that

where m is a positive integer. It follows from (3.4) and (3.5) that

By the Taylor expansion we have

where \(0<\phi _{k}<1\) for \(k=0,1,\ldots,m-1\). When \(m\geq 3\), substituting (3.11) and (3.12) into (3.10) yields

where

Taking \(u=k+j\), we obtain

Since

we have

Therefore,

Because of (2.8) and (3.3) it follows that

where

Observe that

which implies that

if

According to (2.8) and (3.3), (3.20) is equivalent to

Thus, if \(m=2q+1\) and \(N\geq -\frac{c_{m+1}(z)}{c_{m}(z)}\), then

As a consequence, by (3.15) we have

Similarly, if \(m=2p+2\) and \(N\geq -\frac{c_{m+1}(z)}{c_{m}(z)}\), then

which implies

This completes the proof. □

The coefficients \(c_{k}(z)\) in Theorem 3.1 are computed explicitly. In the following theorem, we provide an alternative approach to compute the coefficients recursively.

Theorem 3.2

The coefficients \(c_{k}(z)\) in Theorem 3.1 can be determined by the following recurrence relation:

Proof

From (3.3) and (2.9) it is easy to verify that

For \(k\geq 3\), by (2.7), (2.8), and (3.3), we calculate the right-hand side of (3.23):

Thus, the proof is complete. □

Remark 3.1

By the above recurrence relation, we obtain the first few cases of \(c_{k}(z)\):

Observing the above, we find the following properties for \(c_{k}(z)\).

Theorem 3.3

For \(0< z<1\), the coefficients \(c_{k}(z)\) in Theorem 3.1 satisfy

In particular, if \(\frac{1}{7}< z<1\), then

Proof

By (2.8), we have

which implies that (3.24) is true because \(b_{j}(z)>0\) for all \(j\geq 0\) and \(0< z<1\). By (3.26), we further have

For \(1\leq j\leq k-2\),

Thus,

From (2.8) it follows that

By (3.29) and (3.30), we obtain

It is easy to verify that when \(1/7< z<1\) there holds

which implies that (3.25) is true. □

The following theorem gives the conditions for N in a simpler form than those in Theorem 3.1.

Theorem 3.4

Let p and q be any positive integers. If \(N\geq (2p+2) (\frac{(p+1)z}{1-z}+ \frac{1}{3} )\), then

If \(N\geq (2q+1) ((q+\frac{1}{2})\frac{z}{1-z}+\frac{1}{3} )\), then

Proof

By (2.8) we have

If \(N\geq m (\frac{1}{3}+\frac{m}{2}\frac{z}{1-z} )\), we have

for all \(k=0,1,\ldots,m-2\). This implies (3.19) is true. The proof left is similar to that of Theorem 3.1. □

Remark 3.2

In Theorems 3.1 and 3.4, the conditions for N are sufficient but not necessary. It is possible to relax the conditions for N. Even we conjecture that, for some fixed \(z\in (0,1)\), Theorem 3.1 is true for any positive integer N.

Remark 3.3

In a nutshell, according to Theorems 3.1 and 3.4, for sufficiently large N and any positive integers p and q, we have

In particular, when \(z=1/2\), there holds

which generalizes the result (1.11) due to Chen and Han [5] for large N. In view of (3.36), a variety of simple inequalities for \(\gamma (\frac{1}{2} )-\gamma _{N-1} (\frac{1}{2} )\) are derived for sufficiently large N by choosing different parameters p and q.

Remark 3.4

The inequalities for \(\gamma (\frac{1}{t} )-\gamma _{N-1} (\frac{1}{t} )\) were investigated by Ma and Chen [14]. In particular, some inequalities for the cases \(t=3\) and \(t=4\) were presented, see [14, 15, 29]. As examples, here we apply Theorem 3.1 to obtain several simple and new inequalities related to \(\gamma (\frac{1}{3} )- \gamma _{N-1} (\frac{1}{3} )\) and \(\gamma (\frac{1}{4} )- \gamma _{N-1} (\frac{1}{4} )\) for sufficiently large N.

Case \(t=3\):

Case \(t=4\):

Theorem 3.5

Let \(m\geq 3\) be a positive integer. For \(0< z<1\), we have the following asymptotic expansion:

where \(c_{k}(z)\) are described as in (3.3).

Proof

By (3.15) we immediately obtain (3.37). □

Remark 3.5

When \(z=1/2\), we recover the result of Chen and Han [5] about asymptotic expansion of \(\gamma (\frac{1}{2} )- \gamma _{N-1} (\frac{1}{2} )\):

as \(N\rightarrow \infty \).

Sondow and Hadjicostas [25] generalized Somos’ quadratic recurrence constant (1.3) as

Then they established the relation between the generalized Somos’ quadratic recurrence constant \(\sigma _{t}\) and the function \(\gamma (\frac{1}{t} )\):

Recently, Coffey [6] obtained the following integral and series representations for \(\ln \sigma _{t}\):

and

in terms of the polylogarithm function. Using (4.2) and Theorem 3.1, it is natural that we give an estimate for the generalized Somos’ quadratic recurrence constant \(\sigma _{t}\):

where the bounds

Equivalently, (4.5) leads to a general estimate for \(\ln \sigma _{t}\).

Theorem 4.1

Let p and q be any positive integers. For sufficiently large N, we have

where \(c_{k}(z)\) are described as in (3.3).

As a consequence, we easily obtain an asymptotic expansion for \(\ln \sigma _{t}\).

Theorem 4.2

As \(N\rightarrow \infty \), the following asymptotic expansion holds:

where \(m\geq 3\).

Remark 4.1

Theorems 4.1 and 4.2 show that the function

is a good approximation of \(\ln \sigma _{t}\) with the error term \(O (1/(N^{m}t^{N}) )\). In particular, for large t, the approximation effect is great.

When \(t=2\), we have the following estimate and asymptotic expansion for lnσ.

Corollary 4.1

Let p and q be any positive integers. For sufficiently large N, we have

Corollary 4.2

As \(N\rightarrow \infty \), the following asymptotic expansion holds:

where \(m\geq 3\).

In fact, according to Definition (4.1) and Lemma 2.1 or (2.12), we find an alternative approach to get the estimate for \(\ln \sigma _{t}\).

Theorem 4.3

Let N be any positive integer. Then, for \(t>1\) and \(m\geq 1\), we have

where \(0<\kappa <1\) and \(\lambda _{n}(z)=\sum_{k=1}^{n}(\ln k)z^{k}\).

Proof

Taking \(f(x)=\ln x\) and \(z=1/t\) in (2.12), we immediately obtain (4.12). □

It should be noted that the number N in Theorem 4.3 is not restricted except N is a positive integer. By Theorem 4.3, it is clear that the following corollary is true.

Corollary 4.3

For \(t>1\) and \(m\geq 1\), we have

as \(N\rightarrow \infty \).

Also, we can obtain a new type of approximation for \(\ln \sigma _{t}\) similar to Theorem 3.1.

Corollary 4.4

For any positive integers p, q, and N, there holds

In particular, when \(t=2\) we have

and

as \(N\rightarrow \infty \).

Finally, we get a new estimate for \(\gamma (z)\) when \(0< z<1\).

Corollary 4.5

For any positive integers p, q, and N, there holds

Proof

Using relation (4.2) and Corollary 4.3 and replacing t by \(1/z\), we have the desired result. □

In a similar manner, we obtain the following asymptotic expansion for \(\gamma (z)\).

Corollary 4.6

Let \(m\geq 1\) be a positive integer. For \(0< z<1\), we have

as \(N\rightarrow \infty \).

Remark 4.2

Unlike Theorem 3.1, the number N in (4.17) is not restricted except N is a positive integer. In other words, identity (4.17) is true for all \(N\geq 1\).

Remark 4.3

If we approximate \(\gamma (z)\) by the function \(z^{N-2}\ln N-\frac{1-z}{z ^{2}}\lambda _{N-1}(z)-\frac{\ln (1-z)}{z}\) as \(N\rightarrow \infty \), then we arrive at the error bound \(O(z^{N-2}/N)\). By Theorem 3.5, we conclude that it is better that we approximate \(\gamma (z)\) by its partial sum \(\gamma _{N-1}(z)\) because the error bound is \(O(z^{N-1}/N ^{2})\).

The author sincerely appreciates the anonymous referees for their very helpful and detailed comments, which have significantly improved the presentation of this paper.

Funding

This work was supported by the Natural Science Foundation of Zhejiang Province (Grant No. LY18A010001) and the Ningbo Natural Science Foundation (Grant No. 2017A610140).

Affiliations

1. Institute of Mathematics, Zhejiang Wanli University, Ningbo, China

   • Aimin Xu

Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Corresponding author

Correspondence to Aimin Xu.

Competing interests

The author declares that they have no competing interests.

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