The Experts below are selected from a list of 327 Experts worldwide ranked by ideXlab platform

Hana Funkova - One of the best experts on this subject based on the ideXlab platform.

Ondřej Doslý - One of the best experts on this subject based on the ideXlab platform.

Kim Min-soo - One of the best experts on this subject based on the ideXlab platform.

  • Infinite order Linear Differential Equation satisfied by $p$-adic Hurwitz-type Euler zeta functions
    2021
    Co-Authors: Kim Min-soo
    Abstract:

    In 1900, at the international congress of mathematicians, Hilbert claimed that the Riemann zeta function $\zeta(s)$ is not the solution of any algebraic ordinary Differential Equations on its region of analyticity. In 2015, Van Gorder considered the question of whether $\zeta(s)$ satisfies a non-algebraic Differential Equation and showed that it formally satisfies an infinite order Linear Differential Equation. Recently, Prado and Klinger-Logan extended Van Gorder's result to show that the Hurwitz zeta function $\zeta(s,a)$ is also formally satisfies a similar Differential Equation \begin{Equation*}\label{HurDE} T\left[\zeta (s,a) - \frac{1}{a^s}\right] = \frac{1}{(s-1)a^{s-1}}. \end{Equation*} But unfortunately in the same paper they proved that the operator $T$ applied to Hurwitz zeta function $\zeta(s,a)$ does not converge at any point in the complex plane $\mathbb{C}$. In this paper, by defining $T_{p}^{a}$, a $p$-adic analogue of Van Gorder's operator $T,$ we establish an analogue of Prado and Klinger-Logan's Differential Equation satisfied by $\zeta_{p,E}(s,a)$ which is the $p$-adic analogue of the Hurwitz-type Euler zeta functions \begin{Equation*}\label{HEZ} \zeta_E(s,a)=\sum_{n=0}^\infty\frac{(-1)^n}{(n+a)^s}. \end{Equation*} In contrast with the complex case, due to the non-archimedean property, the operator $T_{p}^{a}$ applied to the $p$-adic Hurwitz-type Euler zeta function $\zeta_{p,E}(s,a)$ is convergent $p$-adically in the area of $s\in\mathbb{Z}_{p}$ with $s\neq 1$ and $a\in K$ with $|a|_{p}>1,$ where $K$ is any finite extension of $\mathbb{Q}_{p}$ with ramification index over $\mathbb{Q}_{p}$ less than $p-1.$Comment: 18 pages. Final version. Dedicated to the memory of Prof. David Goss (1952-2017

  • Infinite order Linear Differential Equation satisfied by $p$-adic Hurwitz-type Euler zeta functions
    2020
    Co-Authors: Kim Min-soo
    Abstract:

    In 1900, at the international congress of mathematicians, Hilbert claimed that the Riemann zeta function $\zeta(s)$ is not the solution of any algebraic ordinary Differential Equations on its region of analyticity. In 2015, Van Gorder considered the question of whether $\zeta(s)$ satisfies a non-algebraic Differential Equation and showed that it formally satisfies an infinite order Linear Differential Equation. Recently, Prado and Klinger-Logan extended Van Gorder's result to show that the Hurwitz zeta function $\zeta(s,a)$ is also formally satisfies a similar Differential Equation \begin{Equation*}\label{HurDE} T\left[\zeta (s,a) - \frac{1}{a^s}\right] = \frac{1}{(s-1)a^{s-1}}. \end{Equation*} But unfortunately in the same paper they proved that the operator $T$ applied to Hurwitz zeta function $\zeta(s,a)$ does not converge at any point in the complex plane $\mathbb{C}$. In this paper, by defining $T_{p}^{a}$, a $p$-adic analogue of Van Gorder's operator $T,$ we establish an analogue of Prado and Klinger-Logan's Differential Equation satisfied by $\zeta_{p,E}(s,a)$ which is the $p$-adic analogue of the Hurwitz-type Euler zeta functions \begin{Equation*}\label{HEZ} \zeta_E(s,a)=\sum_{n=0}^\infty\frac{(-1)^n}{(n+a)^s}. \end{Equation*} In contrast with the complex case, due to the non-archimedean property, the operator $T_{p}^{a}$ applied to the $p$-adic Hurwitz-type Euler zeta function $\zeta_{p,E}(s,a)$ is convergent $p$-adically in the area of $s\in\mathbb{Z}_{p}$ with $s\neq 1$ and $a\in K$ with $|a|_{p}>1,$ where $K$ is any finite extension of $\mathbb{Q}_{p}$ with ramification index over $\mathbb{Q}_{p}$ less than $p-1.$Comment: 16 pages, revised version. We would like to thank Professor Lawrence C. Washington for pointing out a gap in the proof of Lemma 3.1 of the original manuscript and for his helpful suggestion

Ondřej Došlý - One of the best experts on this subject based on the ideXlab platform.

Albert Schneider - One of the best experts on this subject based on the ideXlab platform.

  • perturbations of the half Linear euler Differential Equation
    Results in Mathematics, 2000
    Co-Authors: A Elbert, Albert Schneider
    Abstract:

    Oscillation/nonoscillation properties of the perturbed half-Linear Euler Differential Equation in the critical case are investigated. Strong connections are found between these half-Linear Differential Equations and some Linear Differential Equations, whose coefficient is the perturbation itself. In addition if the solutions of the corresponding Linear Differential Equation satisfy two integral inequalities, then the asymptotic form of the solutions of the half-Linear Differential Equation is established. Examples are given for the latter case.