Log-behavior of Two Sequences Related to the Elliptic Integrals
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Acta MathemaƟcae Applicatae Sinica, English Series The Editorial Office of AMAS & Springer-Verlag GmbH Germany 2020
Log-behavior of Two Sequences Related to the Elliptic Integrals Brian Yi SUN1,† , James Jing-Yu ZHAO2 1 Department
of Mathematics and System Science, Xinjiang University, Urumqi 830046, China
(E-mail: [email protected]) 2 School of Mathematics, Tianjin University, Tianjin 300350, China
Abstract
Two interesting sequences arose in the study of the series expansions of the complete elliptic
integrals, which are called the Catalan-Larcombe-French sequence {Pn }n≥0 and the Fennessey-Larcombe-French sequence {Vn }n≥0 respectively. In this paper, we first establish some criteria for determining log-behavior of a sequence based on its three-term recurrence. Then we prove the log-convexity of {Vn2 − Vn−1 Vn+1 }n≥2 and {n!Vn }n≥1 , the ratio log-concavity of {Pn }n≥0 and the sequence {An }n≥0 of Ap´ ery numbers, and the ratio log-convexity of {Vn }n≥1 . Keywords the Catalan-Larcombe-French sequence; the Fennessey-Larcombe-French sequence; Ap´ ery numbers; log-concave; log-convex; three-term recurrence 2000 MR Subject Classification
1
05A20; 11B37; 11B83
Introduction
Recently, there is a rising interest in the study of the log-behavior of the following two sequences defined by n2 Pn = 8(3n2 − 3n + 1)Pn−1 − 128(n − 1)2 Pn−2 ,
(1.1)
(n − 1)n Vn = 8(n − 1)(3n − n − 1)Vn−1 − 128(n − 2)n Vn−2 , 2
2
2
(1.2)
with the initial values P0 = V0 = 1 and P1 = V1 = 8. The sequences {Pn }n≥0 and {Vn }n≥0 are known as the Catalan-Larcombe-French sequence and the Fennessey-Larcombe-French sequence, respectively. The numbers Pn arise naturally from a series expansion of the complete elliptic integral of the first kind, precisely, ( )n √ ∫ π/2 ∞ 1 π ∑ 1 − 1 − c2 √ dθ = Pn . 2 n=0 16 0 1 − c2 sin2 θ And the numbers Vn appear as coefficients in the series expansion of the complete elliptic integral of the second kind, precisely, ( )n √ √ ∫ π/2 √ ∞ π 1 − c2 ∑ 1 − 1 − c2 2 2 1 − c sin θ dθ = Vn 2 16 0 n=0 (see [3, 8–11]). Manuscript received December 29, 2017. Accepted on January 31, 2020. This work was partially supported by the National Science Foundation of Xinjiang Uygur Autonomous Region (No. 2017D01C084) and the National Science Foundation of China (Nos. 11771330 and 11701491). † Corresponding author.
Log-behavior of Two Sequences Related to the Elliptic Integrals
591
Ap´ery[1] introduced the numbers An , i.e., An =
)2 n ( )2 ( ∑ n n+k k=0
k
k
,
which paly a key role in his proof of the irrationality of ζ(3) =
∞ ∑
1/n3 .
n=1
A recurrence relation was also given by Ap´ery, that is, n3 An = (34n3 − 51n2 + 27n − 5)An−1 − (n − 1)3 An−2 ,
(1.3)
with A0 = 1 and A1 = 5. The main objective of this paper is to prove the log-convexity of {Vn2 − Vn−1 Vn+1 }n≥2 and {n!Vn }n≥1 , the ratio log-concavity of {Pn }n≥0 and {An }n≥0 , and the ratio log-convexity of {Vn }n≥1 . Let us first review some background. Recall that a real sequence {Sn }n≥0 is said to be log-concave (resp. log-convex) if Sn2 ≥ Sn−1 Sn+1 (resp. Sn2 ≤ S
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