Inequalities and asymptotic expansions associated with the Wallis sequence
© Lin et al.; licensee Springer. 2014
Received: 12 March 2014
Accepted: 5 June 2014
Published: 18 July 2014
We present the asymptotic expansions of functions involving the ratio of gamma functions and provide formulas for determining the coefficients of the asymptotic expansions. As consequences, we obtain the asymptotic expansions of the Wallis sequence. Also, we establish inequalities for the Wallis sequence.
MSC:40A05, 33B15, 41A60, 26D15.
Several elementary proofs of (1.2) can be found (see, for example, [2–4]). An interesting geometric construction produces (1.2) . Many formulas exist for the representation of π, and a collection of these formulas is listed in [6, 7]. For more on the history of π see [1, 8–10].
Clearly, . The second aim of this paper is to establish inequalities for the Wallis sequence .
2 A useful lemma
The logarithmic derivative of , denoted by , is called psi (or digamma) function, and () are called polygamma functions.
The following lemma is required in our present investigation.
Lemma 1 ([, Corollary 2.1])
where are the Bernoulli numbers.
In Section 4, the proofs of Theorems 3 and 4 make use of inequality (2.2).
3 Asymptotic expansions
Note that the Bernoulli numbers (for ) are defined by (3.2) for .
with the coefficients given by (1.5). From (3.5) and (3.6), we retrieve (1.8).
Even though as many coefficients as we please on the right-hand side of (3.11) can be obtained by using Mathematica, here we aim at giving a formula for determining these coefficients. In fact, Theorem 1 below presents a general asymptotic expansion for which includes (3.11) as its special case.
The proof of Theorem 1 is complete. □
Theorem 1 gives an explicit formula for determining the coefficients of the asymptotic expansion (3.12). Theorem 2 below provides a recurrence relation for determining the coefficients of the asymptotic expansion (3.12).
where (for ) are given in (3.9).
and (3.17) follows. The proof of Theorem 2 is complete. □
In this section, we establish inequalities for the Wallis sequence .
where is a polynomial of degree k with non-negative integer coefficients. In what follows, has the same understanding.
which means that the first inequality in (4.1) is valid for .
which means that the second inequality in (4.1) is valid for . The proof of Theorem 3 is complete. □
We propose the following.
The first inequality holds for , while the second inequality is valid for all .
This proves the claim.
which means that the first inequality in (4.2) is valid for .
This proves the claim.
which means that the second inequality in (4.2) is valid for . The proof of Theorem 4 is complete. □
The authors would like to thank the referees for their careful reading of the manuscript and insightful comments.
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