Skip to content

Advertisement

Open Access

Padé approximant related to asymptotics for the gamma function

Journal of Inequalities and Applications20172017:53

https://doi.org/10.1186/s13660-017-1315-1

Received: 30 November 2016

Accepted: 5 February 2017

Published: 28 February 2017

Abstract

Based on the Padé approximation method, we determine the coefficients \(a_{j}\) and \(b_{j}\) (\(1\leq j\leq k\)) such that
$$ \frac{\Gamma(x+1)}{\sqrt{2\pi x}(x/e)^{x}}=\frac{x^{k}+a_{1}x^{k-1} +\cdots+a_{k}}{x^{k}+b_{1}x^{k-1}+\cdots+b_{k}}+O \biggl(\frac {1}{x^{2k+1}} \biggr),\quad x\to \infty, $$
where \(k\geq1\) is any given integer. Based on the obtained result, we establish new bounds for the gamma function.

Keywords

gamma functionpsi functioninequalityapproximation

MSC

33B1541A6026D15

1 Introduction

Stirling’s formula
$$\begin{aligned} n!\sim\sqrt{2\pi n} \biggl(\frac{n}{e} \biggr)^{n},\quad n \in \mathbb{N}:=\{1,2,\ldots\} \end{aligned}$$
(1.1)
has many applications in statistical physics, probability theory and number theory. Actually, it was first discovered in 1733 by the French mathematician Abraham de Moivre (1667-1754) in the form
$$\begin{aligned} n!\sim\mbox{constant}\cdot\sqrt{n}(n/e)^{n} \end{aligned}$$
when he was studying the Gaussian distribution and the central limit theorem. Afterwards, the Scottish mathematician James Stirling (1692-1770) found the missing constant \(\sqrt{2\pi}\) when he was trying to give the normal approximation of the binomial distribution.
Stirling’s series for the gamma function is given (see [1], p.257, Eq. (6.1.40)) by
$$\begin{aligned} \Gamma(x+1)&\sim\sqrt{2\pi x} \biggl(\frac{x}{e} \biggr)^{x}\exp \Biggl(\sum_{m=1}^{\infty} \frac {B_{2m}}{2m(2m-1)x^{2m-1}} \Biggr) \end{aligned}$$
(1.2)
as \(x\to\infty\), where \(B_{n}\) \((n \in\mathbb{N}_{0}:=\mathbb{N} \cup\{0\} )\) are the Bernoulli numbers defined by the following generating function:
$$ \frac{z}{e^{z}-1}=\sum_{n=0}^{\infty} B_{n} \frac{z^{n}}{n!}, \quad|z|< 2\pi. $$
(1.3)
The following asymptotic formula is due to Laplace:
$$\begin{aligned} \Gamma(x+1)&\sim\sqrt{2\pi x} \biggl(\frac{x}{e} \biggr)^{x} \biggl(1+\frac{1}{12x}+\frac {1}{288x^{2}}- \frac{139}{51{,}840x^{3}}-\frac{571}{2{,}488{,}320x^{4}}+\cdots \biggr) \end{aligned}$$
(1.4)
as \(x\to\infty\) (see [1], p.257, Eq. (6.1.37)).

The expression (1.4) is sometimes incorrectly called Stirling’s series (see [2], pp.2-3). Stirling’s formula is in fact the first approximation to the asymptotic formula (1.4). Stirling’s formula has attracted much interest of many mathematicians and has motivated a large number of research papers concerning various generalizations and improvements (see [354] and the references cited therein). It is interesting to note that the aforementioned mathematicians represent many nationalities. So the topic is of interest for mathematicians from diverse cultural background.

Using the Maple software, we find, as \(x\to\infty\),
$$\begin{aligned} \frac{\Gamma(x+1)}{\sqrt{2\pi x}(x/e)^{x}}=\frac{x+\frac{1}{24}}{x-\frac {1}{24}}+O \biggl(\frac{1}{x^{3}} \biggr) \end{aligned}$$
(1.5)
and
$$\begin{aligned} \frac{\Gamma(x+1)}{\sqrt{2\pi x}(x/e)^{x}}=\frac{x^{2}+\frac{1}{24}x+\frac {293}{8{,}640}}{x^{2}-\frac{1}{24}x+\frac{293}{8{,}640}}+O \biggl(\frac {1}{x^{5}} \biggr). \end{aligned}$$
(1.6)
Based on the Padé approximation method, in this paper we develop the approximation formulas (1.5) and (1.6) to produce a general result. More precisely, we determine the coefficients \(a_{j}\) and \(b_{j}\) (\(1\leq j\leq k\)) such that
$$ \frac{\Gamma(x+1)}{\sqrt{2\pi x}(x/e)^{x}}=\frac{x^{k}+a_{1}x^{k-1}+\cdots +a_{k}}{x^{k}+b_{1}x^{k-1}+\cdots+b_{k}}+O \biggl(\frac{1}{x^{2k+1}} \biggr), \quad x\to \infty, $$
where \(k\geq1\) is any given integer. Based on the obtained result, we establish new bounds for the gamma function.

The numerical values given in this paper have been calculated via the computer program MAPLE 13.

2 Lemmas

The following lemmas are required in our present investigation.

Lemma 2.1

[9]

Let r be a given nonzero real number. The gamma function has the following asymptotic formula:
$$\begin{aligned} \Gamma(x+1)\sim\sqrt{2\pi x} \biggl(\frac{x}{e} \biggr)^{x} \Biggl(1+\sum_{j=1}^{\infty} \frac {b_{j}}{x^{j}} \Biggr)^{1/r},\quad x\to\infty, \end{aligned}$$
(2.1)
with the coefficients \(b_{j}=b_{j}(r) \) (\(j=1,2,\ldots\)) given by
$$ b_{j}=\sum_{k_{1}+2k_{2}+\cdots+jk_{j}=j} \frac{r^{k_{1}+k_{2}+\cdots +k_{j}}}{k_{1}!k_{2}!\cdots k_{j}!} \biggl(\frac{B_{2}}{1\cdot2} \biggr)^{k_{1}} \biggl( \frac{B_{3}}{2\cdot 3} \biggr)^{k_{2}}\cdots \biggl(\frac{B_{j+1}}{j(j+1)} \biggr)^{k_{j}}, $$
(2.2)
where \(B_{n}\) \((n \in\mathbb{N}_{0}:=\mathbb{N} \cup\{0\} )\) are the Bernoulli numbers defined in (1.3), summed over all nonnegative integers \(k_{j}\) satisfying the equation \(k_{1}+2k_{2}+\cdots+jk_{j}=j\).
Laplace formula (1.4) can be rewritten as
$$\begin{aligned} \Gamma(x+1)&\sim\sqrt{2\pi x} \biggl(\frac{x}{e} \biggr)^{x} \Biggl(\sum_{j=0}^{\infty} \frac {c_{j}}{x^{j}} \Biggr),\quad x \to\infty, \end{aligned}$$
(2.3)
with the coefficients \(c_{j}\) given by
$$ \begin{aligned} &c_{0}=1, \\ &c_{j}=\sum _{k_{1}+2k_{2}+\cdots+{j}k_{j}={j}}\frac {1}{k_{1}!k_{2}!\cdots k_{j}!} \biggl(\frac{B_{2}}{1\cdot2} \biggr)^{k_{1}} \biggl(\frac{B_{3}}{2\cdot 3} \biggr)^{k_{2}}\cdots \biggl(\frac{B_{{j}+1}}{{j}({j}+1)} \biggr)^{k_{j}} \quad\mbox{for } j\geq1. \end{aligned} $$
(2.4)

Lemma 2.2

[55], Theorem 8

Let \(n \geq0\) be an integer. The functions
$$ F_{n}(x)=\ln\Gamma(x)- \biggl(x-\frac{1}{2} \biggr)\ln x+x-\frac{1}{2}\ln(2\pi)-\sum_{j=1}^{2n} \frac{B_{2j}}{2j(2j-1)x^{2j-1}} $$
and
$$ G_{n}(x)=-\ln\Gamma(x)+ \biggl(x-\frac{1}{2} \biggr)\ln x-x+\frac{1}{2}\ln(2\pi)+\sum_{j=1}^{2n+1} \frac{B_{2j}}{2j(2j-1)x^{2j-1}} $$
are completely monotonic on \((0, \infty)\). Here \(B_{n}\) \((n \in\mathbb{N}_{0}:=\mathbb{N}\cup\{0\})\) are the Bernoulli numbers.

Remark 2.1

Lemma 2.2 can be stated as follows: for every \(m\in\mathbb{N}_{0}\), the function
$$ R_{m}(x)=(-1)^{m} \Biggl[\ln\Gamma(x)- \biggl(x-\frac{1}{2} \biggr)\ln x+x-\ln \sqrt{2\pi}-\sum _{j=1}^{m}\frac{B_{2j}}{2j(2j-1)x^{2j-1}} \Biggr] $$
is completely monotonic on \((0, \infty)\).

In 2006, Koumandos [56] presented a simpler proof of complete monotonicity of the functions \(R_{m}(x)\). In 2009, Koumandos and Pedersen [57], Theorem 2.1, strengthened this result.

From \(F_{n}'(x)<0\) and \(G_{n}'(x)<0\) for \(x>0\), we obtain
$$ \sum_{j=1}^{2n} \frac{B_{2j}}{2jx^{2j}}< \ln x-\psi(x)-\frac{1}{2x}< \sum _{j=1}^{2n+1}\frac{B_{2j}}{2jx^{2j}}, \quad x>0, $$
(2.5)
where \(\psi(x)=\Gamma'(x)/\Gamma(x)\) is the psi (or digamma) function. Noting that
$$ \psi(x+1)=\psi(x)+\frac{1}{x} $$
holds, we obtain from (2.5) that for \(x>0\),
$$\begin{aligned} &{-}\frac{1}{12x^{2}}+\frac{1}{120x^{4}}-\frac{1}{252x^{6}}+ \frac {1}{240x^{8}}-\frac{1}{132x^{10}}< \psi(x+1)-\ln x-\frac{1}{2x} \\ &\quad < -\frac{1}{12x^{2}}+\frac{1}{120x^{4}}-\frac {1}{252x^{6}}+\frac{1}{240x^{8}}- \frac{1}{132x^{10}}+\frac{691}{32{,}760x^{12}}. \end{aligned}$$
(2.6)

3 Approximations to the gamma function

For our later use, we introduce Padé approximant (see [5861]). Let f be a formal power series
$$\begin{aligned} f(t)=c_{0}+c_{1}t+c_{2}t^{2}+ \cdots. \end{aligned}$$
(3.1)
The Padé approximation of order \((p, q)\) of the function f is the rational function, denoted by
$$\begin{aligned}{} [p/q]_{f}(t)=\frac{\sum_{j=0}^{p}a_{j}t^{j}}{1+\sum_{j=1}^{q}b_{j}t^{j}}, \end{aligned}$$
(3.2)
where \(p\geq0\) and \(q\geq1\) are two given integers, the coefficients \(a_{j}\) and \(b_{j}\) are given by (see [5860])
$$\begin{aligned} \textstyle\begin{cases} a_{0}=c_{0},\\ a_{1}=c_{0}b_{1}+c_{1},\\ a_{2}=c_{0}b_{2}+c_{1}b_{1}+c_{2}\,\ \vdots\\ a_{p} = c_{0}b_{p}+\cdots+ c_{p-1}b_{1} + c_{p},\\ 0 = c_{p+1} + c_{p}b_{1} + \cdots+ c_{p-q+1}b_{q},\\ \vdots&\\ 0 = c_{p+q} + c_{p+q-1}b_{1} + \cdots+ c_{p}b_{q}, \end{cases}\displaystyle \end{aligned}$$
(3.3)
and the following holds:
$$\begin{aligned}{} [p/q]_{f}(t)- f (t) = O\bigl(t^{p+q+1} \bigr). \end{aligned}$$
(3.4)
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 [61])
$$ [p/q]_{f}(t)= \frac{\left| {\scriptsize\begin{matrix}{}t^{q}f_{p-q}(t) & t^{q-1}f_{p-q+1}(t) &\cdots &f_{p}(t) \cr c_{p-q+1} & c_{p-q+2} &\cdots &c_{p+1} \cr \vdots &\vdots &\ddots &\vdots \cr c_{p} & c_{p+1} &\cdots &c_{p+q} \end{matrix}}\right|}{\left| {\scriptsize\begin{matrix}{} t^{q} & t^{q-1} &\cdots &1 \cr c_{p-q+1} & c_{p-q+2} &\cdots &c_{p+1} \cr \vdots &\vdots &\ddots &\vdots \cr c_{p} & c_{p+1} &\cdots &c_{p+q} \end{matrix}}\right|}, $$
(3.5)
with \(f_{n}(x) = c_{0}+ c_{1}x+ \cdots+ c_{n}x^{n}\), the nth partial sum of the series f in (3.1) (\(f_{n}\) is identically zero for \(n < 0\)).
Let
$$\begin{aligned} f(x)=\frac{\Gamma(x+1)}{\sqrt{2\pi x}(x/e)^{x}}. \end{aligned}$$
(3.6)
It follows from (2.3) that, as \(x\to\infty\),
$$\begin{aligned} f(x)\sim\sum_{j=0}^{\infty} \frac{c_{j}}{x^{j}}=1+\frac{1}{12x}+\frac {1}{288x^{2}}-\frac{139}{51{,}840x^{3}}- \frac{571}{2{,}488{,}320x^{4}}+\cdots, \end{aligned}$$
(3.7)
with the coefficients \(c_{j}\) given by (2.4). In what follows, the function f is given in (3.6).
Based on the Padé approximation method, we now give a derivation of formula (1.5). To this end, we consider
$$\begin{aligned}{} [1/1]_{f}(x)=\frac{\sum_{j=0}^{1}a_{j}x^{-j}}{1+\sum_{j=1}^{1}b_{j}x^{-j}}. \end{aligned}$$
Noting that
$$\begin{aligned} c_{0}=1,\qquad c_{1}=\frac{1}{12},\qquad c_{2}=\frac{1}{288},\qquad c_{3}=-\frac {139}{51{,}840},\qquad c_{4}=-\frac{571}{2{,}488{,}320} \end{aligned}$$
(3.8)
holds, we have, by (3.3),
$$\begin{aligned} \textstyle\begin{cases} a_{0}=1,\\ a_{1}=b_{1}+\frac{1}{12},\\ 0 =\frac{1}{288}+ \frac{1}{12}b_{1}, \end{cases}\displaystyle \end{aligned}$$
that is,
$$\begin{aligned} a_{0}=1,\qquad a_{1}=\frac{1}{24},\qquad b_{1}=- \frac{1}{24}. \end{aligned}$$
We thus obtain that
$$ [1/1]_{f}(x)= \frac{1+\frac{1}{24x}}{1-\frac{1}{24x}}=\frac{x+\frac {1}{24}}{x-\frac{1}{24}}, $$
(3.9)
and we have, by (3.4),
$$ \frac{\Gamma(x+1)}{\sqrt{2\pi x}(x/e)^{x}}=\frac{x+\frac{1}{24}}{x-\frac {1}{24}}+O \biggl(\frac{1}{x^{3}} \biggr). $$
We now give a derivation of formula (1.6). To this end, we consider
$$\begin{aligned}{} [2/2]_{f}(x)=\frac{\sum_{j=0}^{2}a_{j}x^{-j}}{1+\sum_{j=1}^{2}b_{j}x^{-j}}. \end{aligned}$$
Noting that (3.8) holds, we have, by (3.3),
$$\begin{aligned} \textstyle\begin{cases} a_{0}=1,\\ a_{1}=b_{1}+\frac{1}{12},\\ a_{2} =b_{2}+\frac{1}{12}b_{1}+ \frac{1}{288}, \\ 0 = -\frac{139}{51{,}840}+\frac{1}{288}b_{1}+\frac{1}{12}b_{2},\\ 0 = -\frac{571}{2{,}488{,}320}-\frac{139}{51{,}840}b_{1}+\frac{1}{288}b_{2}, \end{cases}\displaystyle \end{aligned}$$
that is,
$$\begin{aligned} a_{0}=1,\qquad a_{1}=\frac{1}{24},\qquad a_{2} = \frac{293}{8{,}640},\qquad b_{1}=-\frac{1}{24},\qquad b_{2}= \frac{293}{8{,}640}. \end{aligned}$$
We thus obtain that
$$ [2/2]_{f}(x)= \frac{1+\frac{1}{24x}+\frac{293}{8{,}640x^{2}}}{1-\frac {1}{24x}+\frac{293}{8{,}640x^{2}}}=\frac{x^{2}+\frac{1}{24}x+\frac {293}{8{,}640}}{x^{2}-\frac{1}{24}x+\frac{293}{8{,}640}} , $$
(3.10)
and we have, by (3.4),
$$ \frac{\Gamma(x+1)}{\sqrt{2\pi x}(x/e)^{x}}=\frac{x^{2}+\frac{1}{24}x+\frac {293}{8{,}640}}{x^{2}-\frac{1}{24}x+\frac{293}{8{,}640}}+O \biggl(\frac {1}{x^{5}} \biggr). $$

From the Padé approximation method and the expansion (3.7), we now present a general result given by Theorem 3.1.

Theorem 3.1

The Padé approximation of order \((p, q)\) of the Laplace asymptotic formula of the function \(f(x)=\frac{\Gamma(x+1)}{\sqrt{2\pi x}(x/e)^{x}}\) (at the point \(x=\infty\)) is the following rational function:
$$\begin{aligned}{} [p/q]_{f}(x)=\frac{1+\sum_{j=1}^{p}a_{j}x^{-j}}{1+\sum_{j=1}^{q}b_{j}x^{-j}}=x^{q-p} \biggl(\frac{x^{p}+a_{1}x^{p-1}+\cdots +a_{p}}{x^{q}+b_{1}x^{q-1}+\cdots+b_{q}} \biggr), \end{aligned}$$
(3.11)
where \(p\geq1\) and \(q\geq1\) are any given integers, the coefficients \(a_{j}\) and \(b_{j}\) are given by
$$\begin{aligned} \textstyle\begin{cases} a_{1}=b_{1}+c_{1},\\ a_{2}=b_{2}+c_{1}b_{1}+c_{2},\\ \vdots\\ a_{p} = b_{p}+\cdots+ c_{p-1}b_{1} + c_{p},\\ 0 = c_{p+1} + c_{p}b_{1} + \cdots+ c_{p-q+1}b_{q},\\ \vdots&\\ 0 = c_{p+q} + c_{p+q-1}b_{1} + \cdots+ c_{p}b_{q}, \end{cases}\displaystyle \end{aligned}$$
(3.12)
and \(c_{j}\) is given in (2.4), and the following holds:
$$\begin{aligned} f (x) -[p/q]_{f}(x) = O \biggl(\frac{1}{x^{p+q+1}} \biggr),\quad x\to \infty. \end{aligned}$$
(3.13)
Moreover, we have
$$ [p/q]_{f}(x)= \frac{\left| {\scriptsize\begin{matrix}{}\frac{1}{x^{q}}f_{p-q}(x) & \frac{1}{x^{q-1}}f_{p-q+1}(x) &\cdots &f_{p}(x) \cr c_{p-q+1} & c_{p-q+2} &\cdots &c_{p+1} \cr \vdots &\vdots &\ddots &\vdots \cr c_{p} & c_{p+1} &\cdots &c_{p+q} \end{matrix}}\right|}{\left| {\scriptsize\begin{matrix}{} \frac{1}{x^{q}} & \frac{1}{x^{q-1}} &\cdots &1 \cr c_{p-q+1} & c_{p-q+2} &\cdots &c_{p+1} \cr \vdots &\vdots &\ddots &\vdots \cr c_{p} & c_{p+1} &\cdots &c_{p+q} \end{matrix}}\right|}, $$
(3.14)
with \(f_{n}(x)=\sum_{j=0}^{n}\frac{c_{j}}{x^{j}}\), the nth partial sum of the asymptotic series (3.7).

Remark 3.1

Using (3.14), we can also derive (3.9) and (3.10). Indeed, we have
$$ [1/1]_{f}(x)=\frac{ \left| {\scriptsize\begin{matrix}{} \frac{1}{x}f_{0}(x) &f_{1}(x) \cr c_{1} &c_{2} \end{matrix}}\right|}{\left| {\scriptsize\begin{matrix}{} \frac{1}{x} &1 \cr c_{1} &c_{2} \end{matrix}}\right|} = \frac{\left| {\scriptsize\begin{matrix}{} \frac{1}{x} &1+\frac{1}{12x}\cr \frac{1}{12} &\frac{1}{288} \end{matrix}}\right|}{\left| {\scriptsize\begin{matrix}{} \frac{1}{x} &1 \cr \frac{1}{12} &\frac{1}{288} \end{matrix}}\right|}=\frac{x+\frac{1}{24}}{x-\frac{1}{24}} $$
and
$$\begin{aligned}{} [2/2]_{f}(x)&=\frac{ \left| {\scriptsize\begin{matrix}{} \frac{1}{x^{2}}f_{0}(x) & \frac{1}{x}f_{1}(x) &f_{2}(x) \cr c_{1} &c_{2} &c_{3} \cr c_{2} &c_{3} &c_{4} \end{matrix}}\right|}{\left| {\scriptsize\begin{matrix}{} \frac{1}{x^{2}} &\frac{1}{x} &1 \cr c_{1} &c_{2} &c_{3} \cr c_{2} &c_{3} &c_{4} \end{matrix}}\right|} = \frac{\left| {\scriptsize\begin{matrix}{} \frac{1}{x^{2}} & \frac{1}{x} (1+\frac{1}{12x} ) &1+\frac {1}{12x}+\frac{1}{288x^{2}}\cr \frac{1}{12} &\frac{1}{288} &-\frac{139}{51{,}840} \cr \frac{1}{288} &-\frac{139}{51{,}840} &-\frac{571}{2{,}488{,}320} \end{matrix}}\right|}{\left| {\scriptsize\begin{matrix}{} \frac{1}{x^{2}} &\frac{1}{x} &1 \cr \frac{1}{12} &\frac{1}{288} &-\frac{139}{51{,}840} \cr \frac{1}{288} &-\frac{139}{51{,}840} &-\frac{571}{2{,}488{,}320} \end{matrix}}\right|} =\frac{x^{2}+\frac{1}{24}x+\frac{293}{8{,}640}}{x^{2}-\frac{1}{24}x+\frac{293}{8{,}640}}. \end{aligned}$$

Setting \((p, q)=(k, k)\) in (3.13), we obtain the following corollary.

Corollary 3.1

As \(x\to\infty\),
$$ \frac{\Gamma(x+1)}{\sqrt{2\pi x}(x/e)^{x}}=\frac{x^{k}+a_{1}x^{k-1}+\cdots +a_{k}}{x^{k}+b_{1}x^{k-1}+\cdots+b_{k}}+O \biggl(\frac{1}{x^{2k+1}} \biggr), $$
(3.15)
where \(k\geq1\) is any given integer, the coefficients \(a_{j}\) and \(b_{j}\) (\(1\leq j\leq k\)) are given by
$$\begin{aligned} \textstyle\begin{cases} a_{1}=b_{1}+c_{1},\\ a_{2}=b_{2}+c_{1}b_{1}+c_{2},\\ \vdots\\ a_{k} = b_{k}+\cdots+ c_{k-1}b_{1} + c_{k},\\ 0 = c_{k+1} + c_{k}b_{1} + \cdots+ c_{1}b_{k},\\ \vdots\\ 0 = c_{2k} + c_{2k-1}b_{1} + \cdots+ c_{k}b_{k}, \end{cases}\displaystyle \end{aligned}$$
(3.16)
and \(c_{j}\) is given in (2.4).
Setting \(k=3\) and \(k=4\) in (3.15), respectively, yields
$$\begin{aligned} \frac{\Gamma(x+1)}{\sqrt{2\pi x}(x/e)^{x}}=\frac{x^{3}+\frac{1}{24}x^{2}+\frac {166{,}903}{590{,}688}x+\frac{4{,}406{,}147}{425{,}295{,}360}}{x^{3}-\frac{1}{24}x^{2}+\frac {166{,}903}{590{,}688}x-\frac{4{,}406{,}147}{425{,}295{,}360}}+O \biggl(\frac{1}{x^{7}} \biggr) \end{aligned}$$
(3.17)
and
$$\begin{aligned} \frac{\Gamma(x+1)}{\sqrt{2\pi x}(x/e)^{x}} =\frac{x^{4}+\frac{1}{24}x^{3}+\frac {685{,}893{,}605}{845{,}980{,}224}x^{2}+\frac{14{,}787{,}105{,}577}{456{,}829{,}320{,}960}x+\frac {2{,}749{,}505{,}046{,}083}{153{,}494{,}651{,}842{,}560}}{x^{4}-\frac{1}{24}x^{3}+\frac {685{,}893{,}605}{845{,}980{,}224}x^{2}-\frac{14{,}787{,}105{,}577}{456{,}829{,}320{,}960}x+\frac {2{,}749{,}505{,}046{,}083}{153{,}494{,}651{,}842{,}560}}+O \biggl(\frac{1}{x^{9}} \biggr). \end{aligned}$$
(3.18)

In view of (1.5), (1.6), (3.17) and (3.18), we pose the following conjecture.

Conjecture 3.1

The coefficients \(a_{j}\) and \(b_{j}\) (\(1\leq j\leq k\)) in (3.15) satisfy the following relation:
$$ a_{j}=(-1)^{j}b_{j}, \quad j=1,2,\ldots, k. $$
(3.19)

4 Inequalities for the gamma function

Formulas (3.17) and (3.18) motivate us to establish the following theorem.

Theorem 4.1

The following inequalities hold:
$$\begin{aligned} U(x)< \frac{\Gamma(x+1)}{\sqrt{2\pi x}(x/e)^{x}}< V(x), \end{aligned}$$
(4.1)
where
$$\begin{aligned} U(x)=\frac{x^{4}+\frac{1}{24}x^{3}+\frac{685{,}893{,}605}{845{,}980{,}224}x^{2}+\frac {14{,}787{,}105{,}577}{456{,}829{,}320{,}960}x+\frac {2{,}749{,}505{,}046{,}083}{153{,}494{,}651{,}842{,}560}}{x^{4}-\frac{1}{24}x^{3}+\frac {685{,}893{,}605}{845{,}980{,}224}x^{2}-\frac{14{,}787{,}105{,}577}{456{,}829{,}320{,}960}x+\frac {2{,}749{,}505{,}046{,}083}{153{,}494{,}651{,}842{,}560}} \end{aligned}$$
(4.2)
and
$$\begin{aligned} V(x)=\frac{x^{3}+\frac{1}{24}x^{2}+\frac{166{,}903}{590{,}688}x+\frac {4{,}406{,}147}{425{,}295{,}360}}{x^{3}-\frac{1}{24}x^{2}+\frac{166{,}903}{590{,}688}x-\frac {4{,}406{,}147}{425{,}295{,}360}}. \end{aligned}$$
(4.3)
The left-hand side inequality holds for \(x\geq3\), while the right-hand side inequality is valid for \(x\geq2\).

Proof

It suffices to show that
$$\begin{aligned} F(x)>0 \quad\mbox{for } x\geq3 \quad\mbox{and}\quad G(x)< 0 \quad\mbox{for } x\geq2, \end{aligned}$$
where
$$\begin{aligned} F(x)=\ln\Gamma(x+1)- \biggl(x+\frac{1}{2} \biggr)\ln x+x-\ln\sqrt{2\pi }- \ln U(x) \end{aligned}$$
and
$$\begin{aligned} G(x)=\ln\Gamma(x+1)- \biggl(x+\frac{1}{2} \biggr)\ln x+x-\ln\sqrt{2\pi }- \ln V(x). \end{aligned}$$
Differentiating \(F(x)\) and applying the second inequality in (2.6) yield
$$\begin{aligned} F'(x)&=\psi(x+1)-\ln x-\frac{1}{2x}-\frac{U'(x)}{U(x)} \\ &< -\frac{1}{12x^{2}}+\frac{1}{120x^{4}}-\frac{1}{252x^{6}}+\frac {1}{240x^{8}}- \frac{1}{132x^{10}}+\frac{691}{32{,}760x^{12}}-\frac {U'(x)}{U(x)} \\ &=-\frac{P_{10}(x-3)}{720{,}720x^{12}P_{8}(x)}, \end{aligned}$$
where
$$\begin{aligned} P_{10}(x)={}&1{,}698{,}313{,}885{,}002{,}591{,}369{,}403{,}376{,}359{,}237{,}155{,}137 \\ &{}+7{,}041{,}090{,}100{,}510{,}955{,}203{,}400{,}650{,}726{,}407{,}309{,}444x \\ &{}+12{,}215{,}302{,}599{,}727{,}743{,}342{,}615{,}877{,}184{,}100{,}329{,}802x^{2} \\ &{}+12{,}025{,}928{,}200{,}234{,}176{,}519{,}514{,}968{,}711{,}811{,}967{,}964x^{3} \\ &{}+7{,}551{,}739{,}592{,}924{,}831{,}815{,}437{,}063{,}682{,}435{,}942{,}293x^{4} \\ &{}+3{,}187{,}338{,}342{,}726{,}357{,}084{,}428{,}868{,}676{,}747{,}628{,}952x^{5} \\ &{}+920{,}408{,}575{,}975{,}851{,}494{,}996{,}412{,}447{,}435{,}781{,}084x^{6} \\ &{}+180{,}133{,}255{,}608{,}389{,}118{,}267{,}365{,}601{,}710{,}648{,}784x^{7} \\ &{}+22{,}910{,}271{,}532{,}985{,}226{,}283{,}927{,}122{,}357{,}066{,}246x^{8} \\ &{}+1{,}711{,}635{,}468{,}441{,}001{,}446{,}976{,}320{,}994{,}717{,}320x^{9} \\ &{}+57{,}054{,}515{,}614{,}700{,}048{,}232{,}544{,}033{,}157{,}244x^{10} \end{aligned}$$
and
$$\begin{aligned} P_{8}(x)={}&\bigl(153{,}494{,}651{,}842{,}560x^{4}+6{,}395{,}610{,}493{,}440x^{3}\\ &{}+124{,}448{,}535{,}691{,}200x^{2}+4{,}968{,}467{,}473{,}872x \\ &{} +2{,}749{,}505{,}046{,}083\bigr) \bigl(153{,}494{,}651{,}842{,}560x^{4}\\ &{}-6{,}395{,}610{,}493{,}440x^{3}+124{,}448{,}535{,}691{,}200x^{2}\\ &{} -4{,}968{,}467{,}473{,}872x+2{,}749{,}505{,}046{,}083\bigr). \end{aligned}$$
Hence, \(F'(x)<0\) for \(x\geq3\), and we have
$$\begin{aligned} F(x)>\lim_{t\to\infty}F(t)=0 \quad\mbox{for } x\geq3. \end{aligned}$$
Differentiating \(G(x)\) and applying the first inequality in (2.6) yield
$$\begin{aligned}[b] G'(x)&=\psi(x+1)-\ln x-\frac{1}{2x}-\frac{V'(x)}{V(x)} \\ &>-\frac{1}{12x^{2}}+\frac{1}{120x^{4}}-\frac{1}{252x^{6}}+\frac {1}{240x^{8}}- \frac{1}{132x^{10}}-\frac{V'(x)}{V(x)} \\ &=\frac{Q_{8}(x-2)}{55{,}440x^{10}Q_{6}(x)}, \end{aligned} $$
where
$$\begin{aligned} Q_{8}(x)={}&2{,}456{,}573{,}428{,}493{,}290{,}077{,}832 +14{,}719{,}278{,}306{,}954{,}453{,}533{,}828x\\ &{}+32{,}394{,}299{,}960{,}322{,}640{,}776{,}801x^{2} \\ &{}+37{,}478{,}643{,}384{,}199{,}534{,}772{,}000x^{3}+25{,}805{,}343{,}259{,}499{,}481{,}612{,}340x^{4} \\ &{}+11{,}004{,}898{,}939{,}796{,}249{,}295{,}384x^{5}+2{,}862{,}385{,}365{,}338{,}807{,}176{,}962x^{6} \\ &{}+416{,}852{,}240{,}076{,}239{,}943{,}360x^{7}+26{,}053{,}265{,}004{,}764{,}996{,}460x^{8} \end{aligned}$$
and
$$\begin{aligned} Q_{6}(x)={}&\bigl(425{,}295{,}360x^{3}+17{,}720{,}640x^{2}+120{,}170{,}160x+4{,}406{,}147 \bigr) \\ &{} \times\bigl(425{,}295{,}360x^{3}-17{,}720{,}640x^{2}+120{,}170{,}160x-4{,}406{,}147 \bigr). \end{aligned}$$
Hence, \(G'(x)>0\) for \(x\geq2\), and we have
$$\begin{aligned} G(x)< \lim_{t\to\infty}G(t)=0 \quad\mbox{for } x\geq2. \end{aligned}$$
The proof is complete. □

Remark 4.1

Following the same method as the one used in the proof of Theorem 4.1, we can prove the double inequality
$$\begin{aligned} \frac{x^{2}+\frac{1}{24}x+\frac{293}{8{,}640}}{x^{2}-\frac{1}{24}x+\frac {293}{8{,}640}}< \frac{\Gamma(x+1)}{\sqrt{2\pi x}(x/e)^{x}}< \frac{x+\frac {1}{24}}{x-\frac{1}{24}} \end{aligned}$$
(4.4)
for \(x\geq2\). We here omit it. Some computer experiments indicate that inequalities (4.1) and (4.4) are valid for \(x\geq1\).

In view of (4.1) and (4.4), we pose the following conjecture.

Conjecture 4.1

If k is odd, then for \(x\geq1\),
$$ \frac{\Gamma(x+1)}{\sqrt{2\pi x}(x/e)^{x}}< \frac{x^{k}+a_{1}x^{k-1}+\cdots +a_{k}}{x^{k}+b_{1}x^{k-1}+\cdots+b_{k}}, $$
(4.5)
where the coefficients \(a_{j}\) and \(b_{j}\) (\(1\leq j \leq k\)) are determined in (3.16). If k is even, then inequality (4.5) is reversed.

5 Comparison

In 2011, Mortici [47] showed by numerical computations that his formula
$$ \begin{aligned} n!\sim\sqrt{2\pi n} \biggl( \frac{n}{e} \biggr)^{n}\exp \biggl(\frac{1}{12n+\frac{2}{5n}} \biggr)= \mu_{n} \end{aligned} $$
(5.1)
is much stronger than other known formulas such as:
$$\begin{aligned} &n!\sim\sqrt{2\pi} \biggl(\frac{n+1/2}{e} \biggr)^{n+1/2}= \beta_{n}\quad (\mbox{Burnside [8]}), \end{aligned}$$
(5.2)
$$\begin{aligned} &n!\sim\frac{\sqrt{2\pi}e^{-n}n^{n+1}}{\sqrt{n-1/6}}=\delta_{n}\quad (\mbox{Batir [4]}), \end{aligned}$$
(5.3)
$$\begin{aligned} &n!\sim\sqrt{2\pi \biggl(n+\frac{1}{6} \biggr)} \biggl(\frac{n}{e} \biggr)^{n}=\gamma_{n}\quad(\mbox{Gosper [19]}), \end{aligned}$$
(5.4)
$$\begin{aligned} &n!\sim\sqrt{\pi} \biggl(\frac{n}{e} \biggr)^{n} \biggl(8n^{3}+4n^{2}+n+\frac {1}{30} \biggr)^{1/6}=\rho_{n}\quad (\mbox{Ramanujan [62], p.339}). \end{aligned}$$
(5.5)
In 2012, Mahmoud et al. [31] showed numerically that their formula
$$ n!\sim\sqrt{2\pi n} \biggl( \frac{n}{e} \biggr)^{n}\exp \biggl(\frac{1}{20n}+ \frac{1}{30}\zeta(2, n+1/2) \biggr)=\lambda_{n} $$
(5.6)
has a superiority over Mortici’s formula (5.1). Here \(\zeta(s,a)\) denotes the Hurwitz (or generalized) zeta function defined by
$$ \zeta(s, a):=\sum_{k=0}^{\infty} \frac{1}{(k+a)^{s}} \quad\bigl(\Re(s)>1; a\notin Z_{0}^{-}\bigr), $$
\(Z_{0}^{-}\) being the set of nonpositive integers.
From (3.17) and (3.18), we obtain
$$\begin{aligned} n!\sim\sqrt{2\pi n} \biggl(\frac{n}{e} \biggr)^{n} \frac{n^{3}+\frac {1}{24}n^{2}+\frac{166{,}903}{590{,}688}n+\frac{4{,}406{,}147}{425{,}295{,}360}}{n^{3}-\frac {1}{24}n^{2}+\frac{166{,}903}{590{,}688}n-\frac{4{,}406{,}147}{425{,}295{,}360}}=U_{n} \end{aligned}$$
(5.7)
and
$$\begin{aligned} n!&\sim\sqrt{2\pi n} \biggl(\frac{n}{e} \biggr)^{n} \frac{n^{4}+\frac {1}{24}n^{3}+\frac{685{,}893{,}605}{845{,}980{,}224}n^{2}+\frac {14{,}787{,}105{,}577}{456{,}829{,}320{,}960}n+\frac {2{,}749{,}505{,}046{,}083}{153{,}494{,}651{,}842{,}560}}{n^{4}-\frac{1}{24}n^{3}+\frac {685{,}893{,}605}{845{,}980{,}224}n^{2}-\frac{14{,}787{,}105{,}577}{456{,}829{,}320{,}960}n+\frac {2{,}749{,}505{,}046{,}083}{153{,}494{,}651{,}842{,}560}} \\ &=V_{n}. \end{aligned}$$
(5.8)
We here offer some numerical computations (see Table 1) to show the superiority of our sequences \((U_{n})_{n\geq1}\) and \((V_{n})_{n\geq1}\) over the sequence \((\lambda_{n})_{n\geq1}\).
Table 1

Comparison among approximation formulas ( 5.6 )-( 5.8 )

n

\(\boldsymbol {\frac{\lambda_{n}-n!}{n!}}\)

\(\boldsymbol {\frac{U_{n}-n!}{n!}}\)

\(\boldsymbol {\frac{n!-V_{n}}{n!}}\)

10

1.7686 × 10−9

3.6355 × 10−11

3.5843 × 10−13

100

1.7855 × 10−14

3.7108 × 10−18

3.7317 × 10−22

1,000

1.7857 × 10−19

3.7115 × 10−25

3.7332 × 10−31

10,000

1.7857 × 10−24

3.7115 × 10−32

3.7333 × 10−40

Declarations

Acknowledgements

The authors thank the referees for helpful comments.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
Department of Mathematics, East China Normal University, Shanghai, People’s Republic of China
(2)
School of Mathematics and Informatics, Henan Polytechnic University, Jiaozuo, China

References

  1. Abramowitz, M, Stegun, IA (eds.): Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series, vol. 55, 9th printing. National Bureau of Standards, Washington (1972) MATHGoogle Scholar
  2. Copson, ET: Asymptotic Expansions. Cambridge University Press, Cambridge (1965) View ArticleMATHGoogle Scholar
  3. Alzer, H: On Ramanujan’s double inequality for the gamma function. Bull. Lond. Math. Soc. 35, 601-607 (2003) MathSciNetView ArticleMATHGoogle Scholar
  4. Batir, N: Sharp inequalities for factorial n. Proyecciones 27, 97-102 (2008) MathSciNetView ArticleMATHGoogle Scholar
  5. Batir, N: Very accurate approximations for the factorial function. J. Math. Inequal. 4, 335-344 (2010) MathSciNetView ArticleMATHGoogle Scholar
  6. Batir, N: Improving Stirling’s formula. Math. Commun. 16, 105-114 (2011) MathSciNetMATHGoogle Scholar
  7. Burić, T, Elezović, N: New asymptotic expansions of the gamma function and improvements of Stirling’s type formulas. J. Comput. Anal. Appl. 13, 785-795 (2011) MathSciNetMATHGoogle Scholar
  8. Burnside, W: A rapidly convergent series for \(\log N!\). Messenger Math. 46, 157-159 (1917) MATHGoogle Scholar
  9. Chen, CP: Unified treatment of several asymptotic formulas for the gamma function. Numer. Algorithms 64, 311-319 (2013) MathSciNetView ArticleMATHGoogle Scholar
  10. Chen, CP: Inequalities and asymptotic expansions associated with the Ramanujan and Nemes formulas for the gamma function. Appl. Math. Comput. 261, 337-350 (2015) MathSciNetGoogle Scholar
  11. Chen, CP: Asymptotic expansions of the gamma function related to Windschitl’s formula. Appl. Math. Comput. 245, 174-180 (2014) MathSciNetMATHGoogle Scholar
  12. Chen, CP, Batir, N: Some inequalities and monotonicity properties associated with the gamma and psi functions. Appl. Math. Comput. 218, 8217-8225 (2012) MathSciNetMATHGoogle Scholar
  13. Chen, CP, Lin, L: Remarks on asymptotic expansions for the gamma function. Appl. Math. Lett. 25, 2322-2326 (2012) MathSciNetView ArticleMATHGoogle Scholar
  14. Chen, CP, Liu, JY: Complete monotonicity properties and asymptotic expansions of the logarithm of the gamma function. Math. Inequal. Appl. 18, 379-388 (2015) MathSciNetMATHGoogle Scholar
  15. Chen, CP, Liu, JY: Inequalities and asymptotic expansions for the gamma function. J. Number Theory 149, 313-326 (2015) MathSciNetView ArticleMATHGoogle Scholar
  16. Chen, CP, Paris, RB: Inequalities, asymptotic expansions and completely monotonic functions related to the gamma function. Appl. Math. Comput. 250, 514-529 (2015) MathSciNetMATHGoogle Scholar
  17. Chen, CP: A sharp version of Ramanujan’s inequality for the factorial function. Ramanujan J. 39, 149-154 (2016) MathSciNetView ArticleMATHGoogle Scholar
  18. Chen, CP: Monotonicity properties, inequalities and asymptotic expansions associated with the gamma function. Appl. Math. Comput. 283, 385-396 (2016) MathSciNetGoogle Scholar
  19. Gosper, RW: Decision procedure for indefinite hypergeometric summation. Proc. Natl. Acad. Sci. USA 75, 40-42 (1978) MathSciNetView ArticleMATHGoogle Scholar
  20. Hirschhorn, MD: A new version of Stirling’s formula. Math. Gaz. 90, 286-291 (2006) View ArticleGoogle Scholar
  21. Hirschhorn, MD, Villarino, MB: A refinement of Ramanujan’s factorial approximation. Ramanujan J. 34, 73-81 (2014) MathSciNetView ArticleMATHGoogle Scholar
  22. Karatsuba, EA: On the asymptotic representation of the Euler gamma function by Ramanujan. J. Comput. Appl. Math. 135, 225-240 (2001) MathSciNetView ArticleMATHGoogle Scholar
  23. Lin, L: On Stirling’s formula remainder. Appl. Math. Comput. 247, 494-500 (2014) MathSciNetMATHGoogle Scholar
  24. Lin, L, Chen, CP: Asymptotic formulas for the gamma function by Gosper. J. Math. Inequal. 9, 541-551 (2015) MathSciNetView ArticleMATHGoogle Scholar
  25. Lu, D: A generated approximation related to Burnside’s formula. J. Number Theory 136, 414-422 (2014) MathSciNetView ArticleMATHGoogle Scholar
  26. Lu, D, Feng, J, Ma, C: A general asymptotic formula of the gamma function based on the Burnside’s formula. J. Number Theory 145, 317-328 (2014) MathSciNetView ArticleMATHGoogle Scholar
  27. Lu, D, Song, L, Ma, C: A generated approximation of the gamma function related to Windschitl’s formula. J. Number Theory 140, 215-225 (2014) MathSciNetView ArticleMATHGoogle Scholar
  28. Lu, D, Song, L, Ma, C: Some new asymptotic approximations of the gamma function based on Nemes’ formula, Ramanujan’s formula and Burnside’s formula. Appl. Math. Comput. 253, 1-7 (2015) MathSciNetMATHGoogle Scholar
  29. Lu, D, Wang, X: A generated approximation related to Gosper’s formula and Ramanujan’s formula. J. Math. Anal. Appl. 406, 287-292 (2013) MathSciNetView ArticleMATHGoogle Scholar
  30. Lu, D, Wang, X: A new asymptotic expansion and some inequalities for the gamma function. J. Number Theory 140, 314-323 (2014) MathSciNetView ArticleMATHGoogle Scholar
  31. Mahmoud, M, Alghamdi, MA, Agarwal, RP: New upper bounds of n!. J. Inequal. Appl. 2012, 27 (2012). http://link.springer.com/article/10.1186/1029-242X-2012-27 MathSciNetView ArticleMATHGoogle Scholar
  32. Mortici, C: An ultimate extremely accurate formula for approximation of the factorial function. Arch. Math. (Basel) 93, 37-45 (2009) MathSciNetView ArticleMATHGoogle Scholar
  33. Mortici, C: Sharp inequalities related to Gosper’s formula. C. R. Math. Acad. Sci. Paris 348, 137-140 (2010) MathSciNetView ArticleMATHGoogle Scholar
  34. Mortici, C: Product approximations via asymptotic integration. Am. Math. Mon. 117, 434-441 (2010) MathSciNetView ArticleMATHGoogle Scholar
  35. Mortici, C: New improvements of the Stirling formula. Appl. Math. Comput. 217, 699-704 (2010) MathSciNetMATHGoogle Scholar
  36. Mortici, C: The asymptotic series of the generalized Stirling formula. Comput. Math. Appl. 60, 786-791 (2010) MathSciNetView ArticleMATHGoogle Scholar
  37. Mortici, C: Asymptotic expansions of the generalized Stirling approximations. Math. Comput. Model. 52, 1867-1868 (2010) MathSciNetView ArticleMATHGoogle Scholar
  38. Mortici, C: A class of integral approximations for the factorial function. Comput. Math. Appl. 59, 2053-2058 (2010) MathSciNetView ArticleMATHGoogle Scholar
  39. Mortici, C: Ramanujan formula for the generalized Stirling approximation. Appl. Math. Comput. 217, 2579-2585 (2010) MathSciNetMATHGoogle Scholar
  40. Mortici, C: Best estimates of the generalized Stirling formula. Appl. Math. Comput. 215, 4044-4048 (2010) MathSciNetMATHGoogle Scholar
  41. Mortici, C: On the gamma function approximation by Burnside. Appl. Math. E-Notes 11, 274-277 (2011) MathSciNetMATHGoogle Scholar
  42. Mortici, C: On Gospers formula for the Gamma function. J. Math. Inequal. 5, 611-614 (2011) MathSciNetView ArticleMATHGoogle Scholar
  43. Mortici, C: Improved asymptotic formulas for the gamma function. Comput. Math. Appl. 61, 3364-3369 (2011) MathSciNetView ArticleMATHGoogle Scholar
  44. Mortici, C: A new fast asymptotic series for the gamma function. Ramanujan J. 38, 549-559 (2015) MathSciNetView ArticleMATHGoogle Scholar
  45. Mortici, C: An improvement of the Ramanujan formula for approximation of the Euler gamma function. Carpath. J. Math. 28, 301-304 (2012) MathSciNetMATHGoogle Scholar
  46. Mortici, C: Ramanujan’s estimate for the gamma function via monotonicity arguments. Ramanujan J. 25, 149-154 (2011) MathSciNetView ArticleMATHGoogle Scholar
  47. Mortici, C: A new Stirling series as continued fraction. Numer. Algorithms 56, 17-26 (2011) MathSciNetView ArticleMATHGoogle Scholar
  48. Mortici, C: On Ramanujan’s large argument formula for the gamma function. Ramanujan J. 26, 185-192 (2011) MathSciNetView ArticleMATHGoogle Scholar
  49. Mortici, C: A continued fraction approximation of the gamma function. J. Math. Anal. Appl. 402, 405-410 (2013) MathSciNetView ArticleMATHGoogle Scholar
  50. Mortici, C: Very accurate estimates of the polygamma functions. Asymptot. Anal. 68, 125-134 (2010) MathSciNetMATHGoogle Scholar
  51. Mortici, C: A new Stirling series as continued fraction. Numer. Algorithms 56, 17-26 (2011) MathSciNetView ArticleMATHGoogle Scholar
  52. Nemes, G: Asymptotic expansion for \(\log n!\) in terms of the reciprocal of a triangular number. Acta Math. Hung. 129, 254-262 (2010) MathSciNetView ArticleMATHGoogle Scholar
  53. Nemes, G: More accurate approximations for the gamma function. Thai J. Math. 9, 21-28 (2011) MathSciNetMATHGoogle Scholar
  54. Nemes, G: New asymptotic expansion for the Gamma function. Arch. Math. (Basel) 95, 161-169 (2010) MathSciNetView ArticleMATHGoogle Scholar
  55. Alzer, H: On some inequalities for the gamma and psi functions. Math. Comput. 66, 373-389 (1997) MathSciNetView ArticleMATHGoogle Scholar
  56. Koumandos, S: Remarks on some completely monotonic functions. J. Math. Anal. Appl. 324, 1458-1461 (2006) MathSciNetView ArticleMATHGoogle Scholar
  57. Koumandos, S, Pedersen, HL: Completely monotonic functions of positive order and asymptotic expansions of the logarithm of Barnes double gamma function and Euler’s gamma function. J. Math. Anal. Appl. 355, 33-40 (2009) MathSciNetView ArticleMATHGoogle Scholar
  58. Bercu, G: Padé approximant related to remarkable inequalities involving trigonometric functions. J. Inequal. Appl. 2016, 99 (2016). http://www.doc88.com/p-0037658479714.html View ArticleMATHGoogle Scholar
  59. Bercu, G: The natural approach of trigonometric inequalities-Pade approximant. J. Math. Inequal. 11, 181-191 (2017) View ArticleMATHGoogle Scholar
  60. Bercu, G, Wu, S: Refinements of certain hyperbolic inequalities via the Padé approximation method. J. Nonlinear Sci. Appl. 9, 5011-5020 (2016) MathSciNetMATHGoogle Scholar
  61. Brezinski, C, Redivo-Zaglia, M: New representations of Padé, Padé-type, and partial Padé approximants. J. Comput. Appl. Math. 284, 69-77 (2015) MathSciNetView ArticleMATHGoogle Scholar
  62. Ramanujan, S: The Lost Notebook and Other Unpublished Papers. Springer, Berlin (1988) MATHGoogle Scholar

Copyright

© The Author(s) 2017

Advertisement