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Twologconvexity of the CatalanLarcombeFrench sequence
Journal of Inequalities and Applications volume 2015, Article number: 404 (2015)
Abstract
The CatalanLarcombeFrench sequence \(\{P_{n}\}_{n\geq0}\) arises in a series expansion of the complete elliptic integral of the first kind. It has been proved that the sequence is logbalanced. In the paper, by exploring a criterion due to Chen and Xia for testing 2logconvexity of a sequence satisfying threeterm recurrence relation, we prove that the new sequence \(\{P^{2}_{n}P_{n1}P_{n+1}\}_{n\geq1}\) are strictly logconvex and hence the CatalanLarcombeFrench sequence is strictly 2logconvex.
Introduction
This paper is concerned with the logbehavior of the CatalanLarcombeFrench sequence. To begin with, let us recall that a sequence \(\{z_{n}\}_{n\geq0}\) is said to be logconcave if
and it is logconvex if
Meanwhile, the sequence \(\{z_{n}\}_{n\geq0}\) is called strictly logconcave (resp. logconvex) if the inequality in (1.1) (resp. (1.2)) is strict for all \(n\geq1\). We call \(\{z_{n}\}_{n\geq0}\) logbalanced if the sequence itself is logconvex while \(\{\frac{z_{n}}{n!}\}_{n\geq0}\) is logconcave.
Given a sequence \(A=\{z_{n}\}_{n\geq0}\), define the operator \(\mathcal {L}\) by
where \(s_{n}=z_{n1}z_{n+1}z_{n}^{2}\) for \(n\geq1\). We say that \(\{z_{n}\} _{n\geq0}\) is klogconvex (resp. klogconcave) if \(\mathcal{L}^{j}(A)\) is logconvex (resp. logconcave) for all \(j=0,1,\ldots,k1\), and that \(A=\{z_{n}\}_{n\geq0}\) is ∞logconvex (resp. ∞logconcave) if \(\mathcal{L}^{k}(A)\) is logconvex (resp. logconcave) for any \(k\geq0\). Similarly, we can define strict klogconcavity or strict klogconvexity of a sequence.
It is worthy to mention that besides that they are fertile sources of inequalities, logconvexity and logconcavity have many applications in some different mathematical disciplines, such as geometry, probability theory, combinatorics, and so on. See the surveys due to Brenti [1] and Stanley [2] for more details. Additionally, it is clear that the logbalancedness implies the logconvexity and a sequence \(\{z_{n}\}_{n\geq0}\) is logconvex (resp. logconcave) if and only if its quotient sequence \(\{\frac {z_{n}}{z_{n1}}\}_{n\geq1}\) is nondecreasing (resp. nonincreasing). It is also known that the quotient sequence of a logbalanced sequence does not grow too fast. Therefore, logbehavior are important properties of combinatorial sequences and they are instrumental in obtaining the growth rate of a sequence. Hence the logbehaviors of a sequence deserves to be investigated.
In this paper, we investigate the 2logbehavior of the CatalanLarcombeFrench sequence, denoted by \(\{P_{n}\}_{n\geq0}\), which arises in connection with series expansions of the complete elliptic integrals of the first kind [3, 4]. To be precise, for \(0<c<1\),
Furthermore, the numbers \(P_{n}\) can be written as the following sum:
see [5], A05317. Besides, the number \(P_{n}\) satisfies threeterm recurrence relations [4] as follows:
with the initial values \(P_{0}=1\) and \(P_{1}=8\).
Recently, Zhao [4] studied the logbehavior of the CatalanLarcombeFrench sequence and proved that the sequence \(\{P_{n}\} _{n\geq0}\) is logbalanced. What is more, the CatalanLarcombeFrench sequence has many interesting properties and the reader can refer [3, 4, 6]. In the sequel, we study the 2logbehavior of the sequences and obtain the following result.
Theorem 1.1
The CatalanLarcombeFrench sequence \(\{P_{n}\}_{n\geq0}\) is strictly 2logconvex, that is,
where \(\mathcal{P}_{n}=P_{n}^{2}P_{n1}P_{n+1}\).
We will give our proof of Theorem 1.1 in the third section by utilizing a testing criterion, which is proposed by Chen and Xia [7].
To make this paper selfcontained, let us recall their criterion.
Theorem 1.2
(Chen and Xia [7])
Suppose \(\{z_{n}\}_{n\geq0}\) is a positive logconvex sequence that satisfies the following threeterm recurrence relation:
Let
and
Assume that \(c_{3}(n)<0\) and \(\Delta(n)\geq0\) for all \(n\geq N\), where N is a positive integer. If there exist \(f_{n}\) and \(g_{n}\) such that, for all \(n\geq N\),

(I)
\(f_{n}\leq\frac{z_{n}}{z_{n1}}\leq g_{n}\);

(II)
\(f_{n}\geq\frac{2c_{2}(n)\sqrt{\Delta(n)}}{6c_{3}(n)}\);

(III)
\(c_{3}(n)g_{n}^{3}+c_{2}(n)g_{n}^{2}+c_{1}(n)g_{n}+c_{0}(n)\geq0\),
then we see that \(\{z_{n}\}_{n\geq N}\) is 2logconvex, that is, for \(n\geq N\),
With respect to the theory in this field, it should be mentioned that the logbehavior of a sequence which satisfies a threeterm recurrence has been extensively studied; see Liu and Wang [8], Chen et al. [9, 10], Liggett [11], Došlić [12], etc.
Bounds for \(\frac{P_{n}}{P_{n1}}\)
Before proving Theorem 1.1, we need the following two lemmas.
Lemma 2.1
Let
and \(P_{n}\) be the sequence defined by the recurrence relation (1.3). Then we have, for all \(n\geq1\),
Proof
We proceed the proof by induction. First note that, for \(n=1\) and \(n=2\), we have \(\frac{P_{1}}{P_{0}}=8>\frac{232}{45}\) and \(\frac{P_{2}}{P_{1}}=10>\frac{464}{60}\). Assume that the inequality (2.1) is valid for \(n\leq k\). We will show that
By the recurrence (1.3), we have
in which the last inequality follows by
for all \(k\geq1\). This completes the proof. □
Lemma 2.2
Let
and \(P_{n}\) be the sequence defined by the recurrence relation (1.3). Then we have, for all \(n\geq6\),
Proof
First note that, for \(n=6\), we have \(\frac{P_{6}}{P_{5}}=\frac{3\text{,}562}{269}< g_{6}=\frac{358}{27}\). Assume that, for \(k\geq6\), the inequality (2.2) is valid for \(n\leq k\). We will show that
By the recurrence (1.3), we have
Consider
for all \(k\geq2\). So we see that, for all \(n\geq6\), the inequality (2.2) holds by induction. □
With the above lemmas in hand, we are now in a position to prove our main result in the next section.
Proof of Theorem 1.1
In this section, by using the criterion of Theorem 1.2, we can show that the CatalanLarcombeFrench sequence is strictly 2logconvex.
To begin with, the following lemma, which is obtained by Zhao [4], is indispensable for us.
Lemma 3.1
(Zhao [4])
The CatalanLarcombeFrench sequence is logbalanced.
By the definition of logbalanced sequence, we know that \(\{P_{n}\} _{n\geq0}\) is logconvex.
Proof of Theorem 1.1
By Lemma 3.1, it suffices for us to show that
According to the recurrence relation (1.3), we see that
By taking \(a(n)\), \(b(n)\) in \(c_{0},\ldots,c_{3}\), we can obtain
for all \(n\geq1\). Besides, we have to verify that, for some positive integer N, the conditions (II) and (III) in Theorem 1.2 hold for all \(n\geq N\). That is,
Let
and
To show (3.1), it is equivalent to show that, for some positive integers N, \(\delta(n)\geq0\) and \(\delta^{2}(n)\geq\Delta(n)\). By calculating, we easily find that, for all \(n\geq1\),
and for all \(n\geq3\),
Thus, take \(N=3\) and, for all \(n\geq N\), we have \(\delta(n)\geq0\), \(\delta^{2}(n)\geq\Delta(n)\), which follows from the inequality (3.1). We show the inequality (3.2) for some positive integer M. Note that, by Lemma 2.2 and some calculations, we have
Take \(M=6\), it is not difficult to verify that, for all \(n\geq M\),
Let \(N_{0}=\max\{N,M\}=6\), then for all \(n\geq6\), all of the above inequalities hold. By Lemma 3.1 and Theorem 1.2, the CatalanLarcombeFrench sequence \(\{P_{n}\}_{n\geq6}\) is strictly 2logconvex for all \(n\geq6\). What is more, one can easily test that these numbers \(\{P_{n}\}_{0\leq n\leq8}\) also satisfy the property of 2logconvexity by simple calculations. Therefore, the whole sequence \(\{P_{n}\}_{n\geq0}\) is strictly 2logconvex. This completes the proof. □
It deserves to be mentioned that by considerable calculations and plenty of verifications, the following conjectures should be true.
Conjecture 3.2
The CatalanLarcombeFrench sequence is ∞logconvex.
Conjecture 3.3
The quotient sequence \(\{\frac{P_{n}}{P_{n1}}\}_{n\geq1}\) of the CatalanLarcombeFrench sequence is logconcave, equivalently, for all \(n\geq2\),
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Acknowledgements
Research supported by NSFC (No. 11161046) and by the Xingjiang Talent Youth Project (No. 2013721012).
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Sun, B.Y., Wu, B. Twologconvexity of the CatalanLarcombeFrench sequence. J Inequal Appl 2015, 404 (2015) doi:10.1186/s1366001509200
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MSC
 05A20
 11B37
 11B83
Keywords
 logbalanced sequence
 logconvex sequence
 logconcave sequence
 the CatalanLarcombeFrench sequence
 threeterm recurrence