Rational approximations to values of Bell polynomials at points involving Euler's constant and zeta values

TL;DR

This paper develops new explicit rational approximations that converge rapidly to specific values of Bell polynomials evaluated at points involving Euler's constant and zeta values, advancing computational methods in this area.

Contribution

It introduces explicit simultaneous rational approximations to Bell polynomial values at special points involving Euler's constant and zeta values, with sub-exponential convergence.

Findings

Convergent rational approximations for Bell polynomial values.

Explicit formulas involving Euler's constant and zeta values.

Sub-exponential convergence rate of the approximations.

Abstract

In this paper, we present new explicit simultaneous rational approximations converging sub-exponentially to the values of Bell polynomials at the points of the form $(\gamma, 1! (2a+1)\zeta(2), 2!\zeta(3),..., (m-1)!(a+1+(-1)^ma)\zeta(m)),$ $m=1,2,...,a,$ $a\in{\mathbb N}.$