Properties and applications of the reciprocal logarithm numbers

Abstract

Via a graphical method, which codes tree diagrams composed of partitions, a novel power series expansion is derived for the reciprocal of the logarithmic function ln∈(1+z), whose coefficients represent an infinite set of fractions. These numbers, which are called reciprocal logarithm numbers and are denoted by A k, converge to zero as k→∞. Several properties of the numbers are then obtained including recursion relations and their relationship with the Stirling numbers of the first kind. Also appearing here are several applications including a new representation for Euler's constant known as Hurst's formula and another for the logarithmic integral. From the properties of the A k it is found t..