- Research
- Open Access
Two-log-convexity of the Catalan-Larcombe-French sequence
- Brian Y Sun^{1}Email author and
- Baoyindureng Wu^{1}
https://doi.org/10.1186/s13660-015-0920-0
© Sun and Wu 2015
- Received: 14 August 2015
- Accepted: 30 November 2015
- Published: 18 December 2015
Abstract
The Catalan-Larcombe-French 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 log-balanced. In the paper, by exploring a criterion due to Chen and Xia for testing 2-log-convexity of a sequence satisfying three-term recurrence relation, we prove that the new sequence \(\{P^{2}_{n}-P_{n-1}P_{n+1}\}_{n\geq1}\) are strictly log-convex and hence the Catalan-Larcombe-French sequence is strictly 2-log-convex.
Keywords
- log-balanced sequence
- log-convex sequence
- log-concave sequence
- the Catalan-Larcombe-French sequence
- three-term recurrence
MSC
- 05A20
- 11B37
- 11B83
1 Introduction
It is worthy to mention that besides that they are fertile sources of inequalities, log-convexity and log-concavity 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 log-balancedness implies the log-convexity and a sequence \(\{z_{n}\}_{n\geq0}\) is log-convex (resp. log-concave) if and only if its quotient sequence \(\{\frac {z_{n}}{z_{n-1}}\}_{n\geq1}\) is nondecreasing (resp. nonincreasing). It is also known that the quotient sequence of a log-balanced sequence does not grow too fast. Therefore, log-behavior are important properties of combinatorial sequences and they are instrumental in obtaining the growth rate of a sequence. Hence the log-behaviors of a sequence deserves to be investigated.
Recently, Zhao [4] studied the log-behavior of the Catalan-Larcombe-French sequence and proved that the sequence \(\{P_{n}\} _{n\geq0}\) is log-balanced. What is more, the Catalan-Larcombe-French sequence has many interesting properties and the reader can refer [3, 4, 6]. In the sequel, we study the 2-log-behavior of the sequences and obtain the following result.
Theorem 1.1
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 self-contained, let us recall their criterion.
Theorem 1.2
(Chen and Xia [7])
- (I)
\(f_{n}\leq\frac{z_{n}}{z_{n-1}}\leq g_{n}\);
- (II)
\(f_{n}\geq\frac{-2c_{2}(n)-\sqrt{\Delta(n)}}{6c_{3}(n)}\);
- (III)
\(c_{3}(n)g_{n}^{3}+c_{2}(n)g_{n}^{2}+c_{1}(n)g_{n}+c_{0}(n)\geq0\),
With respect to the theory in this field, it should be mentioned that the log-behavior of a sequence which satisfies a three-term recurrence has been extensively studied; see Liu and Wang [8], Chen et al. [9, 10], Liggett [11], Došlić [12], etc.
2 Bounds for \(\frac{P_{n}}{P_{n-1}}\)
Before proving Theorem 1.1, we need the following two lemmas.
Lemma 2.1
Proof
Lemma 2.2
Proof
With the above lemmas in hand, we are now in a position to prove our main result in the next section.
3 Proof of Theorem 1.1
In this section, by using the criterion of Theorem 1.2, we can show that the Catalan-Larcombe-French sequence is strictly 2-log-convex.
To begin with, the following lemma, which is obtained by Zhao [4], is indispensable for us.
By the definition of log-balanced sequence, we know that \(\{P_{n}\} _{n\geq0}\) is log-convex.
Proof of Theorem 1.1
It deserves to be mentioned that by considerable calculations and plenty of verifications, the following conjectures should be true.
Conjecture 3.2
The Catalan-Larcombe-French sequence is ∞-log-convex.
Conjecture 3.3
Declarations
Acknowledgements
Research supported by NSFC (No. 11161046) and by the Xingjiang Talent Youth Project (No. 2013721012).
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
References
- Brenti, F: Unimodal, log-concave, and Pólya frequency sequences in combinatorics. Mem. Am. Math. Soc. 413, 1-106 (1989) MathSciNetGoogle Scholar
- Stanley, RP: Log-concave and unimodal sequences in algebra, combinatorics, and geometry. Ann. N.Y. Acad. Sci. 576, 500-535 (1989) View ArticleGoogle Scholar
- Jarvis, F, Verrill, HA: Supercongruences for the Catalan-Larcombe-French numbers. Ramanujan J. 22, 171-186 (2010). doi:10.1007/s11139-009-9218-5 MATHView ArticleMathSciNetGoogle Scholar
- Zhao, F-Z: The log-behavior of the Catalan-Larcombe-French sequences. Int. J. Number Theory 10, 177-182 (2014) View ArticleMathSciNetGoogle Scholar
- Sloane, NJA: The on-line encyclopedia of integer sequences. https://oeis.org/
- Larcombe, PJ, French, DR: On the ‘other’ Catalan numbers: a historical formulation re-examined. Congr. Numer. 143, 33-64 (2000) MATHMathSciNetGoogle Scholar
- Chen, WYC, Xia, EXW: The 2-log-convexity of the Apéry numbers. Proc. Am. Math. Soc. 139, 391-400 (2011) MATHView ArticleMathSciNetGoogle Scholar
- Liu, LL, Wang, Y: On the log-convexity of combinatorial sequences. Adv. Appl. Math. 39, 453-476 (2007) MATHView ArticleGoogle Scholar
- Chen, WYC, Guo, JJF, Wang, LXW: Infinitely log-monotonic combinatorial sequences. Adv. Appl. Math. 52, 99-120 (2014) MATHView ArticleMathSciNetGoogle Scholar
- Chen, WYC, Guo, JJF, Wang, LXW: Zeta functions and the log-behavior of combinatorial sequences. Proc. Edinb. Math. Soc. (2). To appear. arXiv:1208.5213
- Liggett, TM: Ultra logconcave sequences and negative dependence. J. Comb. Theory, Ser. A 79, 315-325 (1997) MATHView ArticleMathSciNetGoogle Scholar
- Došlić, T: Log-balanced combinatorial sequences. Int. J. Math. Math. Sci. 4, 507-522 (2005) Google Scholar