Abstract
Based on the Padé approximation method, in this paper we determine the coefficients \(a_{j}\) and \(b_{j}\) such that
where \(k\geq0\) is any given integer. Based on the obtained result, we establish a more accurate formula for approximating π, which refines some known results.
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1 Introduction
It is well known that the number π satisfies the following inequalities:
where
This result is due to Wallis (see [1]).
Based on a basic theorem in mathematical statistics concerning unbiased estimators with minimum variance, Gurland [1] yielded a closer approximation to π than that afforded by (1.1), namely,
By using (1.2), Brutman [2] and Falaleev [3] established estimates of the Landau constants.
Mortici [4], Theorem 2, improved Gurland’s result (1.2) and obtained the following double inequality:
We see from (1.3) that
Based on the Padé approximation method, in this paper we develop the approximation formula (1.4) to produce a general result. More precisely, we determine the coefficients \(a_{j}\) and \(b_{j}\) such that
where \(k\geq0\) is any given integer. Based on the obtained result, we establish a more accurate formula for approximating π, which refines some known results.
The numerical values given in this paper have been calculated via the computer program MAPLE 13.
2 Lemmas
Euler’s gamma function \(\Gamma(x)\) is one of the most important functions in mathematical analysis and has applications in diverse areas. The logarithmic derivative of \(\Gamma(x)\), denoted by \(\psi(x)=\Gamma'(x)/\Gamma(x)\), is called the psi (or digamma) function.
The following lemmas are required in the sequel.
Lemma 2.1
[5]
Let \(r\neq0\) be a given real number and \(\ell\geq0\) be a given integer. The following asymptotic expansion holds:
with the coefficients \(p_{j}\equiv p_{j}(\ell,r)\ (j\in\mathbb{N})\) given by
where \(B_{j}\) are the Bernoulli numbers summed over all nonnegative integers \(k_{j}\) satisfying the equation
In particular, setting \((\ell, r)=(0, -2)\) in (2.1) yields
where the coefficients \(c_{j}\equiv p_{j}(0, -2)\ (j\in\mathbb{N})\) are given by
summed over all nonnegative integers \(k_{j}\) satisfying the equation
Lemma 2.2
[5]
Let \(m, n\in\mathbb{N}\). Then, for \(x>0\),
In particular, we have
where
and
For our later use, we introduce Padé approximant (see [6–11]). Let f be a formal power series
The Padé approximation of order \((p, q)\) of the function f is the rational function, denoted by
where \(p\geq0\) and \(q\geq1\) are two given integers, the coefficients \(a_{j}\) and \(b_{j}\) are given by (see [6–8, 10, 11])
and the following holds:
Thus, the first \(p + q + 1\) coefficients of the series expansion of \([p/q]_{f}\) are identical to those of f. Moreover, we have (see [9])
with \(f_{n}(x) = c_{0}+ c_{1}x+ \cdots+ c_{n}x^{n}\), the nth partial sum of the series f in (2.7).
3 Main results
Let
It follows from (2.3) that, as \(x\to \infty\),
with the coefficients \(c_{j}\) given by (2.4). In what follows, the function f is given in (3.1).
Based on the Padé approximation method, we now give a derivation of formula (1.4). To this end, we consider
Noting that
holds, we have, by (2.9),
that is,
We thus obtain that
and we have, by (2.10),
Noting that
holds, replacing x by n in (3.4) yields (1.4).
From the Padé approximation method introduced in Section 2 and the asymptotic expansion (3.2), we obtain a general result given by Theorem 3.1. As a consequence, we obtain (1.5).
Theorem 3.1
The Padé approximation of order \((p, q)\) of the asymptotic formula of the function \(f(x)=x (\frac{\Gamma (x+\frac{1}{2} )}{\Gamma (x+1)} )^{2}\) (at the point \(x=\infty\)) is the following rational function:
where \(p\geq0\) and \(q\geq1\) are two given integers and \(q=p+1\) (an empty sum is understood to be zero), the coefficients \(a_{j}\) and \(b_{j}\) are given by
and \(c_{j}\) is given in (2.4), and the following holds:
Moreover, we have
with \(f_{n}(x)=\sum_{j=0}^{n}\frac{c_{j}}{x^{j}}\), the nth partial sum of the asymptotic series (3.2).
Remark 3.1
Using (3.9), we can also derive (3.3). Indeed, we have
Replacing x by n in (3.8) applying (3.5), we obtain the following corollary.
Corollary 3.1
As \(n\to\infty\),
where \(p\geq0\) and \(q\geq1\) are two given integers and \(q=p+1\), and the coefficients \(a_{j}\) and \(b_{j}\) are given by (3.7).
Remark 3.2
Setting \((p, q)=(k, k+1)\) in (3.10) yields (1.5).
Setting
in (3.10), respectively, we find
and
as \(n\to\infty\).
Formulas (3.11) and (3.12) motivate us to establish the following theorem.
Theorem 3.2
The following inequality holds:
The left-hand side inequality holds for \(x\geq4\), while the right-hand side inequality is valid for \(x\geq3\).
Proof
It suffices to show that
where
and
Using the following asymptotic expansion (see [12]):
we obtain that
Differentiating \(F(x)\) and applying the first inequality in (2.6), we find
where
and
Hence, \(F'(x)<0\) for \(x\geq4\), and we have
Differentiating \(G(x)\) and applying the second inequality in (2.6), we find
where
and
Hence, \(G'(x)>0\) for \(x\geq3\), and we have
The proof is complete. □
Corollary 3.2
For \(n \in \mathbb{N}\),
where
and
Proof
Noting that (3.5) holds, we see by (3.13) that the left-hand side of (3.15) holds for \(n\geq4\), while the right-hand side of (3.15) is valid for \(n\geq3\). Elementary calculations show that the left-hand side of (3.15) is also valid for \(n =1, 2\) and 3, and the right-hand side of (3.15) is valid for \(n =1\) and 2. The proof is complete. □
4 Comparison
Recently, Lin [12] improved Mortici’s result (1.3) and obtained the following inequalities:
and
where
Direct computation yields
and
The following numerical computations (see Table 1) would show that \(\delta_{n}< a_{n}\) and \(b_{n}<\omega_{n}\) for \(n\in\mathbb{N}\). That is to say, inequalities (3.15) are sharper than inequalities (4.2).
In fact, we have
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Lin, L., Ma, WC. & Chen, CP. Padé approximant related to the Wallis formula. J Inequal Appl 2017, 132 (2017). https://doi.org/10.1186/s13660-017-1406-z
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DOI: https://doi.org/10.1186/s13660-017-1406-z