Yohei Tachiya

Transcendence of the values of infinite products in several variables

The aim of this paper is to prove the transcendence of certain infinite products. As applications, we get necessary and sufficient conditions for transcendence of the value of % MathType!MTEF!2!1!+-% feaaeaart1ev0aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXanrfitLxBI9gBaerbd9wDYLwzYbItLDharqqt% ubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq% -Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0x% fr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyuam% aaBaaaleaacaaIXaGaaGimaaqabaGccqGH9aqpciGGSbGaaiOBaiaa% ysW7caWGRbWaaSbaaSqaaiaadsfacaaIXaaabeaakiaac+cacaWGRb% WaaSbaaSqaaiaadsfacaaIYaaabeaakiabg2da9iabgkHiTmaabmaa% baGaamyramaaBaaaleaacaWGHbaabeaakiaac+cacaWGsbaacaGLOa% GaayzkaaGaey41aq7aaiWaaeaadaqadaqaaiaadsfadaWgaaWcbaGa% aGOmaaqabaGccqGHsislcaWGubWaaSbaaSqaaiaaigdaaeqaaaGcca% GLOaGaayzkaaGaai4laiaacIcacaWGubWaaSbaaSqaaiaaikdaaeqa% aOGaaGjbVlaadsfadaWgaaWcbaGaamysaaqabaGccaGGPaaacaGL7b% GaayzFaaaaaa!5C4A!

Linear independence of certain Lambert series

\Pi_{k=0}^{\infty}(1+a_{k}^{(1)}{z_{1}r^{k}}+\cdot\cdot\cdot+a_{k}^{(m)}{z_{m}r^{k}})

Pattern sequences in ‹q, r›- numeration systems

at appropriate algebraic points, where r ≥ 2 is an integer and {an(i)}n≥ 0 (1 ≤ i ≤ m) are suitable sequences of algebraic numbers.

Linear independence results for the reciprocal sums of Fibonacci numbers associated with Dirichlet characters

at the rational numbers z = q−1, where d(n) denotes the number of positive divisors of n. Later, Erdős and Graham [8] pointed out that the number ∑∞ n=1 1/(2 −3) was not known to be irrational. Borwein [2], [3] answered this question in the affirmative by proving the irrationality of the numbers ∑∞ n=1 1/(q n − r), where r is a rational number with r = 0, q (n ≥ 1). The second author (see [10]) modified Borwein’s contour-integral constructions from [3] and showed the linear independence of the three numbers

IRRATIONALITY OF LAMBERT SERIES ASSOCIATED WITH A PERIODIC SEQUENCE

Abstract Let q ≥ 2 and 0 ≤ r ≤ q − 2 be integers. In this paper, we study pattern sequences for patterns in ‹ q, r ›-numeration systems through their generating functions. Our result implies that any nontrivial linear combination over ℂ of pattern sequences chosen from different ‹ q, r ›-numeration systems cannot be a linear recurrence sequence. In particular, pattern sequences in different ‹ q, r ›-numeration systems are linearly independent over ℂ, while within one ‹ q, r ›-numeration system they can be linearly dependent ℂ.

Arithmetical properties of infinite products generated by binary recurrences

Let {Fn}n≥0 be the sequence of Fibonacci numbers. The aim of this paper is to give linear independence results over ℚ( 5 ) for the infinite series ∑ n=1 ∞ χ j ( n )/ F n with certain nonprincipal real Dirichlet characters χj. We also deduce the irrationality results for the special principal Dirichlet characters and for other multiplicative functions.

Measure of irrationality for certain infinite series

Let q be an integer with |q| > 1 and {an}n≥1 be an eventually periodic sequence of rational numbers, not identically zero from some point on. Then the number is irrational. In particular, if the periodic sequences of rational numbers are linearly independent over ℚ, then so are the following m + 1 numbers: This generalizes a result of Erdős who treated the case of m = 1 and . The method of proof is based on the original approaches of Chowla and Erdős, together with some results about primes in arithmetic progressions with large moduli of Ahlford, Granville and Pomerance.

Irrationality of Certain Lambert Series

We present arithmetical properties of certain infinite products generated by binary recurrences. Moreover, we give necessary and sufficient conditions for such infinite products to be algebraically independent.

Transcendence of certain infinite products

Let K be either the rational number field Q or an imaginary quadratic field. Let q(|q|>1) be an integer of K, r∈K×(|r|<|q|), and 1⩽l∈Z with qn≠rl(n⩾1). Then we obtain quantitative irrationality results of the number θ = Σn = 1∞rn/(qn−rl).