论文标题
涉及斐波那契数的某些无限产品的代数独立性
Algebraic independence of certain infinite products involving the Fibonacci numbers
论文作者
论文摘要
令$ \ {f_ {n} \} _ {n \ geq0} $为fibonacci数字的序列。本文的目的是给出无限产品\ [\ prod_ {n = 1}^{\ infty} \ left（1+ \ frac {1} {f_ {n}}} \ right），\ qquad \ qquad \ prod_ { 1- \ frac {1} {f_ {n}} \ right）\] \] \] \] \] \]。由此，我们通过将Bertrand的定理应用于Jacobi Theta函数值的代数独立性上，从而推断出上述数字的$ \ mathbb {q} $的代数独立性。
Let $\{F_{n}\}_{n\geq0}$ be the sequence of the Fibonacci numbers. The aim of this paper is to give explicit formulae for the infinite products \[ \prod_{n=1}^{\infty}\left( 1+\frac{1}{F_{n}}\right) ,\qquad\prod_{n=3}^{\infty}\left( 1-\frac{1}{F_{n}}\right) \] in terms of the values of the Jacobi theta functions. From this we deduce the algebraic independence over $\mathbb{Q}$ of the above numbers by applying Bertrand's theorem on the algebraic independence of the values of the Jacobi theta functions.
