## Tsukuba Journal of Mathematics

- Tsukuba J. Math.
- Volume 30, Number 2 (2006), 345-361.

### Algebraic independence of modified reciprocal sums of products of Fibonacci numbers

#### Abstract

In this paper we establish, using Mahler's method, the algebraic independence of reciprocal sums of products of Fibonacci numbers including slowly increasing factors in their numerators (see Theorems 1, 5, and 6 below). Theorems 1 and 4 are proved by using Theorems 2 and 3 stating key formulas of this paper, which are deduced from the crucial Lemma 2. Theorems 5 and 6 are proved by using different technique. From Theorems 2 and 5 we deduce Corollary 2, the algebraic independence of the sum of a certain series and that of its subseries obtained by taking subscripts in a geometric progression.

#### Article information

**Source**

Tsukuba J. Math., Volume 30, Number 2 (2006), 345-361.

**Dates**

First available in Project Euclid: 30 May 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.tkbjm/1496165067

**Digital Object Identifier**

doi:10.21099/tkbjm/1496165067

**Mathematical Reviews number (MathSciNet)**

MR2271304

**Zentralblatt MATH identifier**

1204.11122

#### Citation

Tanaka, Taka-aki. Algebraic independence of modified reciprocal sums of products of Fibonacci numbers. Tsukuba J. Math. 30 (2006), no. 2, 345--361. doi:10.21099/tkbjm/1496165067. https://projecteuclid.org/euclid.tkbjm/1496165067