Generalizing the Reciprocal Logarithm Numbers by Adapting the Partition Method for a Power Series Expansion

Abstract

Recently, a novel method based on the coding of partitions was used to determine a power series expansion for the reciprocal of the logarithmic function, viz. z/ln (1+z). Here we explain how this method can be adapted to obtain power series expansions for other intractable functions. First, the method is adapted to evaluate the Bernoulli numbers and polynomials. As a result, new integral representations and properties are determined for the former. Then via another adaptation of the method we derive a power series expansion for the function z ^s/ln ^s(1+z), whose polynomial coefficients A _k (s) are referred to as the generalized reciprocal logarithm numbers because they reduce to the reciprocal logarithm numbers when s=1. In addition to presenting a general formula for their evaluation, this paper presents various properties of the generalized reciprocal logarithm numbers including general formulas for specific values of s, a recursion relation and a finite sum identity. Other representations in terms of special polynomials are also derived for the A _k (s), which yield general formulas for the highest order coefficients. The paper concludes by deriving new results involving infinite series of the A _k (s) for the Riemann zeta and gamma functions and other mathematical quantities.