Inequalities and asymptotic expansions related to the generalized Somos quadratic recurrence constant.

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.

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 , defined by where the series converges when . Pilehrood and Pilehrood [13] considered the function (). The function generalizes both Euler’s constant and the alternating Euler constant [17, 18].

Sondow and Hadjicostas [15] defined the generalized Somos constant Coffey [19] gave the integral and series representations for : 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 , and Lu and Song [10] improved Mortici’s results and obtained the inequalities: and for .

You and Chen [16] further improved inequalities (1.10)–(1.13). Recently, Chen and Han [7] gave new bounds for : for , and presented the following asymptotic expansion: as . 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 sharper than the one in (1.12), and the upper bound in (1.14) is for sharper than the one in (1.12),

For any positive integer , in this paper we give the asymptotic expansion of as . Based on the result obtained, we establish the inequality for . We also consider the asymptotic expansion for .

Lemmas

Lemma 2.1

As , where is defined by with the coefficients given by the recurrence relation Here, and throughout this paper, an empty sum is understood to be zero.

Proof

Using the Maclaurin series of , we obtain

In view of (2.4), we can let where 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 on both sides of (2.8) yields

For , we obtain , and for , 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 ,

It is well known that for and , which implies that for and , 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 , we have where is defined by with the coefficients given by the recurrence relation

In view of (2.4), we can let where are real numbers to be determined. Write (2.16) as Noting that (2.7) holds, we have Equating coefficients of the term on both sides of (2.17) yields

For , we obtain , and for we have We then obtain the recursive formula (2.15). The proof of Lemma 2.3 is complete. □

The first few coefficients are

Main results

For any positive integer , Theorem 3.1 gives the asymptotic expansion of as .

Theorem 3.1

For any positive integer , we have where is given in (2.2). Namely,

Write (2.1) as where with the coefficients given by the recurrence relation (2.3). From (3.3), we have Adding (3.5) from to , we have which can be written as (3.1). The proof of Theorem 3.1 is complete. □

Remark 3.1

For in (3.2), we obtain (1.15). For in (3.2), we find For in (3.2), we find

Formula (3.7) motivated us to establish Theorem 3.2.

Theorem 3.2

For ,

From the double inequality (2.11), we have where and are given in (2.10). Adding inequalities (3.9) from to , 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 , 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 . 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 , we have with the coefficients given in (2.3).

By using the Maple software, we find, as , 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 , we here provide a pair of recurrence relations for determining the constants and (see Remark 3.4) such that as . 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 , we have as , where and are given by a pair of recurrence relations and with Here are given in (2.3).

In view of (3.13), (3.14), and (3.15), we let where and 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 on the right-hand sides of (3.22) and (3.23), we obtain

Setting and in (3.24), respectively, yields and For , from (3.25) and (3.26) we obtain and for 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 and by using formulas (3.18) and (3.19). This shows how easily we can determine the constants and in (3.17).

Remark 3.4

The constants and in (3.16) are given by

Setting , and 4 in (3.16), respectively, yields (3.13), (3.14), and (3.15).

Noting that holds, Theorem 3.4 presents the asymptotic expansion for .

Theorem 3.4

As , we have where is given in (2.14). Namely,

Write (2.13) as where with the coefficients given by the recurrence relation (2.15).

From (3.29), we have Adding (3.31) from to , 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 has the following expansion:

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 and (see (3.8), (3.10), and (3.11)). The expansion of the alternating Euler constant is also obtained (see (3.33)).