Linear independence results for sums of reciprocals of Fibonacci and Lucas numbers
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LINEAR INDEPENDENCE RESULTS FOR SUMS OF RECIPROCALS OF FIBONACCI AND LUCAS NUMBERS D. DUVERNEY1 , Y. SUZUKI2,∗ and Y. TACHIYA3 1
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Bˆ atiment A1, 110 rue du chevalier Fran¸cais, 59000 Lille, France e-mail: [email protected]
Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya 464-8602, Japan e-mail: [email protected]
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Graduate School of Science and Technology, Hirosaki University, Hirosaki 036-8561, Japan e-mail: [email protected] (Received November 12, 2019; revised April 1, 2020; accepted April 2, 2020)
Abstract. The aim of this paper is to give linear independence results for the values of Lambert type series. As an application, we derive arithmetical properties of the sums of reciprocals of Fibonacci and Lucas numbers associated with certain coprime sequences {nℓ }ℓ≥1 . For example, the three numbers 1 1 , 1, F 2 L 2 p:prime p p:prime p √ are linearly independent over Q( 5), where {Fn } and {Ln } are the Fibonacci and Lucas numbers, respectively.
1. Introduction and results Throughout this paper, let {nℓ }ℓ≥1 be an increasing sequence of positive odd integers satisfying the following two conditions: distinct integers ni and nj are coprime, (H1 ) Any two 1 (H2 ) ∞ is convergent. ℓ=1 nℓ
Example 1.1. It is well known that the ℓ-th prime number pℓ is asymptotically equal to ℓ log ℓ as ℓ → ∞. Hence, the sequence of m-th powers of odd primes {pm ℓ+1 }ℓ≥1 satisfies the conditions (H1 ) and (H2 ) for m ≥ 2. ∗ Corresponding
author. This work was supported by JSPS KAKENHI Grant Numbers JP16J00906 and JP18K03201. Key words and phrases: linear independence, Lambert series, Fibonacci number, Lucas num-
ber. Mathematics Subject Classification: primary 11J72, secondary 11A41. c 2020 0236-5294/$ 20.00 © � 0 Akad´ emiai Kiad´ o, Budapest 0236-5294/$20.00 Akade ´miai Kiado ´, Budapest, Hungary
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D. D. DUVERNEY, DUVERNEY, Y. Y. SUZUKI SUZUKI and and Y. Y. TACHIYA TACHIYA
Example 1.2. The super-prime numbers (also known as prime-indexed primes) are the subsequence of prime numbers that occupy prime-numbered positions within the sequence of all prime numbers. Then the ℓ-th superprime number ppℓ is asymptotically equal to pℓ log pℓ ∼ ℓ(log ℓ)2 as ℓ → ∞, and so the sequence of all super-prime numbers {ppℓ }ℓ≥1 satisfies the conditions (H1 ) and (H2 ). For any positive integer t > 1, Erd˝os [7] showed that the base-t representation of the infinite series ∞ �
(1.1)
ℓ=1
1 tn ℓ − 1
contains arbitrarily long strings of 0 without being identically zero from some point on, and consequently the number (1.1) is irrational. The purpose of this paper is to improve Erd˝os’s method in [7] and give linear independence results for more general series of type (1.1). Let a1 (n) and a3 (n) be the numbers of divisors nℓ of n of the forms 4m + 1 and 4m + 3, respectively. For j = 1, 2, 3, 4, we define fj (z) :=
(1.2)
∞ �
bj (n)z n ,
n=1
where a1 (n) if n ≡ j (mod 4), bj (n) := a3 (n) if n ≡ j + 2 (mod 4), 0 otherwise.
(1.3)
Note that the functions fj (z) (j = 1, 2, 3, 4)
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