Approximation formulas related to Somos' quadratic recurrence constant.

We present two classes of asymptotic expansions related to Somos' quadratic recurrence constant and provide the recursive relations for determining the coefficients of each class of the asymptotic expansions by using Bell polynomials and other techniques. We also present continued fraction approximations related to Somos' quadratic recurrence constant.

Introduction

Somos [1] (see [2, p. 446] and [3]) defined the sequence The first few terms are The following asymptotic expansion is known in the literature: where is known as Somos’ quadratic recurrence constant. Formula (1.1) was proved by Somos, and it is cited in Finch’s book [2, p. 446] as Somos’ result. Note that the coefficient of in Finch’s book is 137, but actually it is incorrect and its correct value is 138 (see Weisstein, Eric W. “Somos’s Quadratic Recurrence Constant.” From MathWorld–A Wolfram Web Resource [3]). The constant σ appears in important problems from pure and applied analysis, it has motivated a large number of research papers (see, for example, [4-17]).

Nemes [15] studied the coefficients in the asymptotic expansion (1.1) and developed recurrence relations. More precisely, Nemes [15, Theorem 1] proved that where the coefficients (for ) are given by the recurrence relation The coefficients also satisfy the following recurrence relation [15, Theorem 3]: where are the ordered Bell numbers defined by the exponential generating function [18, p. 189] The ordered Bell numbers are given explicitly by the formula The first few ordered Bell numbers are Nemes [15, Theorem 2] proved that the generating function of the coefficients has the following representation:

Chen [5, Theorem 2.1] presented a class of asymptotic expansions related to Somos’ quadratic recurrence constant, which includes formula (1.1) as its special case. Let be a given real number. The sequence has the following asymptotic formula: with the coefficients () given by where () denotes the ordered Bell numbers and the summation in (1.8) is taken over all nonnegative integers satisfying the equation .

The first aim of the present paper is to give recursive relations for determining the coefficients in (1.7) (Theorem 2.1). The second aim of the present paper is to establish a more general result, which includes expansion (1.7) as its special case (Theorem 2.2). Our last aim in this paper is to present continued fraction approximations related to Somos’ quadratic recurrence constant (Theorems 3.1 and 3.2).

Asymptotic expansions

Theorem 2.1 below gives recursive relations for determining the coefficients in (1.7) by using the Bell polynomials.

The Bell polynomials, named in honor of Eric Temple Bell, are a triangular array of polynomials given by (see [19, pp. 133–134] and [20, 26]) where the sum is taken over all sequences of nonnegative integers such that The sum is sometimes called the nth complete Bell polynomial. The complete Bell polynomials satisfy the following identity:

In order to contrast them with complete Bell polynomials, the polynomials defined above are sometimes called partial Bell polynomials. The complete Bell polynomials appear in the exponential of a formal power series

The Bell polynomials are quite general polynomials and they have been found in many applications in combinatorics. Comtet [19] devoted much to a thorough presentation of the Bell polynomials in the chapter on identities and expansions. For more results, the reader is referred to [21, Chap. 11] and [22, Chap. 5].

Theorem 2.1

Let r be a given nonzero real number. Then the sequence has the following asymptotic expansion: with the coefficients (for ) given by the recursive relation where (for ) denotes the ordered Bell numbers defined by (1.5).

Proof

From (1.3), it follows that On the other hand, from the definition of , it follows that where (for ) are real numbers to be determined. By using (1.6) and (2.4), we have Therefore it is seen that the in (2.8) can be expressed in terms of the Bell polynomials Bulò et al. [23, Theorem 1] proved that the complete Bell polynomials can be expressed using the following recursive formula: Thus, formula (2.9) can be rewritten as The proof of Theorem 2.1 is complete. □

Remark 2.1

The representation using a recursive algorithm for the coefficients in (1.7) is more practical for numerical evaluation than the expression in (1.8). We can directly calculate in (2.9) by using identity (2.3).

Remark 2.2

We find that a special case of (2.5) when yields immediately the asymptotic formula (1.1). Here, taking and in (2.5), respectively, we give two explicit expressions and

Theorem 2.2 establishes a more general result, which includes Theorem 2.1 as its special case.

Theorem 2.2

Let r be a given nonzero real number and m be a given nonnegative integer. Then the sequence has the following asymptotic expansion: with the coefficients (for ) given by the recursive relation where (for ) denotes the ordered Bell numbers defined by (1.5).

From (2.5), it follows that where (for ) are real numbers to be determined.

Taking the logarithm of (2.14) and applying (1.6) yields Replacing n by x gives Differentiating each side with respect to x yields Hence, and formula (2.13) follows. The proof of Theorem 2.2 is complete. □

Remark 2.3

Setting and in (2.12), respectively, we give two explicit expressions and

Remark 2.4

Setting , we obtain from (2.13) that It is easy to see that (2.6) is equivalent to (2.17). Setting , (2.17) becomes (1.4).

Continued fraction approximations

We define the sequence by We are interested in finding fixed parameters a, b, c, d, p, and q such that converges as fast as possible to zero. This provides the best approximations of the form Our study is based on the following lemma, which is useful for accelerating some convergences, or in constructing some better asymptotic expansions.

Lemma 3.1

([24, 25])

If the sequence converges to zero and if the following limit exists, then where denotes the set of real numbers.

Theorem 3.1

Let the sequence be defined by (3.1). Then, for we have The speed of convergence of the sequence is given by the order estimate as .

First of all, we write the difference as the following power series in : The fastest sequence is obtained when the first six coefficients of this power series vanish. In this case we have Finally, by using Lemma 3.1, we obtain assertion (3.4) of Theorem 3.1. □

Solution (3.3) provides the best approximation of type (3.2),

Now we define the sequence by We are interested in finding fixed parameters , , , , , , , and such that converges as fast as possible to zero. This provides the best approximations of the form Following the same method used in the proof of Theorem 3.1, we can prove Theorem 3.2, we omit it.

Theorem 3.2

Let the sequence be defined by (3.7). Then, for we have and The speed of convergence of the sequence is given by the order estimate as .

Solution (3.9) provides the best approximation of type (3.8)

Conclusions

In this paper, we give asymptotic expansions related to the generalized Somos’ quadratic recurrence constant (Theorems 2.1 and 2.2). We present continued fraction approximations related to Somos’ quadratic recurrence constant (Theorems 3.1 and 3.2).