Properties and Applications of the Reciprocal Logarithm Numbers
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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 that a term of ζ(2) cannot be eliminated by the remaining terms in Hurst’s formula, thereby indicating that Euler’s constant is irrational. Finally, another power series expansion for the reciprocal of arctangent is developed by adapting the preceding material.
KeywordsConvergence Divergent series Equivalence Euler’s constant Harmonic numbers Hurst’s formula Partitions Power series expansions Reciprocal logarithm numbers Recursion relation Regularization Soldner’s constant Stirling numbers of the first kind Tree diagram
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