Little q-Legendre polynomials and irrationality of certain Lambert series

Irrationality proof of certain Lambert series using little q-Jacobi polynomials

We apply the Pade technique to find rational approximations to % \[h^{\pm}(q_1,q_2)=\sum_{k=1}^\infty\frac{\q_1^k}{1\pm \q_2^k}, 0<q_1,q_2<1, q_1\in\mathbb{Q}, q_2=1/p_2, p_2\in\mathbb{N}\setminus\{1\}.\] % A separate section is dedicated to the special case $q_i=q^{r_i}, r_i\in\mathbb{N}, q=1/p, p\in\mathbb{N}\setminus\{1\}$. In this construction we make use of little $q$-Jacobi polynomials. Our rational approximations are good enough to prove the irrationality of $h^{\pm}(q_1,q_2)$ and give an upper bound for the irrationality measure.

关于Ahmes级数的几个无理性问题

On several irrationality problems for Ahmes series

本文利用数学分析与概率论工具研究了单位分数级数的无理性问题，构造了双指数增长序列并推广至多级数情形，同时解决了Erdős、Straus和Stolarsky的相关猜想。