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

Little q-Legendre polynomials and irrationality of certain Lambert series

We show how one can obtain rational approximants for $q$-extensions of the harmonic series and the logarithm (and many other similar quantities) by Padé approximation using little $q$-Legendre polynomials and we show that properties of these orthogonal polynomials indeed prove the irrationality, with an upper bound of the measure of irrationality which is as sharp as the upper bound given by Bundschuh and Väänänen for the harmonic series and a better upper bound than the one given by Matala-aho and Väänänen for the logarithm.

Irrationality proof of a $q$-extension of $ζ(2)$ using little $q$-Jacobi polynomials

We show how one can use Hermite-Padé approximation and little $q$-Jacobi polynomials to construct rational approximants for $ζ_q(2)$. These numbers are $q$-analogues of the well known $ζ(2)$. Here $q=\frac{1}{p}$, with $p$ an integer greater than one. These approximants are good enough to show the irrationality of $ζ_q(2)$ and they allow us to calculate an upper bound for its measure of irrationality: $μ(ζ_q(2))\leq 10π^2/(5π^2-24) \approx 3.8936$. This is sharper than the upper bound given by Zudilin (\textit{On the irrationality measure for a $q$-analogue of $ζ(2)$}, Mat. Sb. \textbf{193} (2…

Irrationality of $ζ_q(1)$ and $ζ_q(2)$

In this paper we show how one can obtain simultaneous rational approximants for $ζ_q(1)$ and $ζ_q(2)$ with a common denominator by means of Hermite-Pade approximation using multiple little q-Jacobi polynomials and we show that properties of these rational approximants prove that 1, $ζ_q(1)$, $ζ_q(2)$ are linearly independent over the rationals. In particular this implies that $ζ_q(1)$ and $ζ_q(2)$ are irrational. Furthermore we give an upper bound for the measure of irrationality.

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。

关于利用Mellin变换生成幂级数、超幂级数、Lambert型级数与Dirichlet型级数族及其若干推论

On the Use of the Mellin Transform to Generate Families of Power, Hyperpower, Lambert and Dirichlet Type Series and Some Consequences

本研究利用Mellin变换处理两类特殊形式的级数，推导其积分表示并进行变换，得到相关积分结果、极限与超幂级数表示，还推广了Riemann关于Zeta、Gamma与Jacobi Theta函数的经典关系式。