Abstract
In this paper, we obtained a simple rational approximation for \({\mathcal {K}}_{a}(r)\):
holds for all \(r\in (0,1),\) where \({\mathcal {K}}_{a}(r)=\) \(\frac{\pi }{2}F\left( a,1-a;1;r^{2}\right) =\frac{\pi }{2}\sum _{n=0}^{\infty }\frac{ \left( a\right) _{n}\left( 1-a\right) _{n}}{(n!)^{2}}r^{2n}\) is the generalized elliptic integral of the first kind, and \(r\mathbf {^{\prime }=} \sqrt{1-r^{2}}\). In particular, when \({\small a}\) is taken as 1/2, 1/3, 1/4 and 1/6 respectively, we can obtain the specific lower bound of the corresponding function.
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1 Introduction
For \(r\in (0,1),\) \(a\in (0,1),\) and \(r^{\prime }=\sqrt{1-r^{2}},\) the generalized elliptic integral of the first kind (see [1, Section 5.5]) is defined by
which can be expressed as a power series
where \(\left( a\right) _{n}\) is the shifted factorial function (or Pochhammer symbol)
In 2000, this special function was rediscovered by [2]. In the particular case \(a=1/2,\) the function \({\mathcal {K}}_{a}(r)\) reduces to \({\mathcal {K}}(r),\) the well-known complete elliptic integral of the first kind:
which is the particular case of the Gaussian hypergeometric function [3,4,5,6,7,8,9]:
Many researchers have obtained some results about this special function \( {\mathcal {K}}_{a}(r)\) ( see [10,11,12,13,14,15,16,17,18,19,20]) . In [21], the upper and lower bounds for \({\mathcal {K}}_{a}(r)\) were shown, and a double inequality was proved as follows:
holds for all \(a\in (0,1/2]\) and \(r\in (0,1)\) if and only if \(\alpha \le \pi /[R(a)\sin (\pi a)]-1\) and \(\beta \ge a\left( 1-a\right) \), where R(x) is the Ramanujan constant function (see [22]).
The aim of this paper is to provide a new concise bound for \({\mathcal {K}} _{a}(r)\) by a rational function of the arugment \(r^{\prime }\) and obtain the following results.
Theorem 1
Let \(a\in (0,1),\) \(r\in (0,1),\) and \(r^{\prime }=\sqrt{1-r^{2}}\). Then
Letting \(a=1/2,1/3,1/4,\) and 1/6 in the above theorem, respectively, we immediately draw the following corollary.
Corollary 1
Let \(r\in (0,1)\) and \(r^{\prime }=\sqrt{1-r^{2}}\). Then
2 Lemmas
In order to prove our main results we need following lemmas.
Lemma 1
Let \(a\in \left( 0,1\right) ,\) \(n\ge 5\). Then
holds.
Proof
The inequality required is equivalent to
We use mathematical induction to prove \(\left( 7\right) \). We can calculate directly
where
The fact that \(g(a)>0\) for all \(a\in \left( 0,1\right) \) can be proved by the following elementary method:
where \(t=\left( a-1/2\right) ^{2}\) and \(0\le t<1/4\). So \(T_{5}>0,\) which implies (7) holds for \(n=5\). Assuming that (7) holds for \(n=m>5\), that is,
where
Next, we prove that (7) is valid for \(n=m+1\). By (9) we have
in order to complete the proof of (7) it suffices to show that
that is
or
In fact,
where
We can prove \(u_{i}(a)>0\) \(\left( i=0,1,2,3\right) \) for all \(a\in \left( 0,1\right) \) as follows:
where \(t=\left( a-1/2\right) ^{2}\) and \(0\le t<1/4\). So \(AD-BC>0\) for all \( m>5\).
This completes the proof of Lemma 1. \(\square \)
Lemma 2
[26,27,28,29,30,31] Let \(\{a_{k}\}_{k=0}^{\infty }\) be a nonnegative real sequence with \(a_{m}>0\) and \(\sum _{k=m+1}^{\infty }a_{k}>0\), and
be a convergent power series on the interval (0, r) \((r>0)\). Then the following statements are true:
- 1.:
-
If \(S(r^{-})\le 0\), then \(S(t)<0\) for all \(t\in (0,r)\);
- 2.:
-
If \(S(r^{-})>0\), then there exists \(t_{0}\in (0,r)\) such that \(S(t)<0\) for \(t\in (0,t_{0})\) and \(S(t)>0\) for \(t\in (t_{0},r)\).
Lemma 2 appeared first in [26], then was proven in [27]. In a recent paper [28], the above power series S(t) in Lemma 2 has be called “NP type power series”.
3 Proof of Theorem 1
By
we have
where
A direct verification yields
for all \(a\in \left( 0,1\right) ,\) and by Lemma 1 we have \(a_{n}<0\) for all \(n\ge 5\).
So no matter what the sign of \(a_{4}\) is, those coefficients in power series of \(-{\mathcal {F}}(r)\) satisfy the conditions of Lemma 2, and \(-{\mathcal {F}} (1^{-})=\infty \). From Lemma 2, it follows that there is a unique \( r_{0}\in \left( 0,1\right) \) such that \(-{\mathcal {F}}(r)<0\) for \(r\in (0,r_{0})\) and \(-{\mathcal {F}}(r)>0\) for \(r\in (r_{0},1)\), that is, \(\mathcal {F }(r)>0\) for \(r\in (0,r_{0})\) and \({\mathcal {F}}(r)<0\) for \(r\in (r_{0},1),\) and \(r_{0}\) is the unique zero of \({\mathcal {F}}(r)\) on \(\left( 0,1\right) \). At the same time, since
and the function
has a unique zero on \(\left( 0,1\right) \), which leads to \(r_{0}=2\sqrt{ a\left( 1-a\right) }/\left( 1+a-a^{2}\right) \). The fact that \(r_{0}\in \left( 0,1\right) \) is true because
and
hold for all \(a\in \left( 0,1\right) \). Clearly, \({\mathcal {S}}(r)>0\) for \( r\in (0,r_{0})\) and \({\mathcal {S}}(r)<0\) for \(r\in (r_{0},1)\). In a word,
holds for all \(r\in \left( 0,1\right) \) with \(r\ne r_{0}\). Because these two functions \({\mathcal {K}}_{a}(r)\) and \({\mathcal {G}}(r\mathbf {^{\prime }})\) are continuous at the point \(r_{0}\), the inequality (2) still holds for \( r=r_{0}\).
The proof of Theorem 1 is complete.
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Acknowledgements
The author is thankful to reviewers for careful corrections to and valuable comments on the original version of this paper. This paper is supported by the Natural Science Foundation of China Grants No. 61772025.
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Zhu, L. A simple rational approximation to the generalized elliptic integral of the first kind. RACSAM 115, 89 (2021). https://doi.org/10.1007/s13398-021-01027-1
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DOI: https://doi.org/10.1007/s13398-021-01027-1