Home > Mathematics, Statistics > Vol 10 No 1 June 2022 Abstract 4
Some New Representations of the Binet's Function Involving Euler Sums
Nikola Naidenov

Abstract
We consider a method for transforming divergent series arising from the Euler - Maclaurin formula into convergent ones. Applying it to the Stirling series forlog??(??)we obtain some new representations of the J. Binet function. In the special case when the argument ?? is integer or half-integer number the formulas take an elegant form involving colored Euler sums. Note that the obtained equalities are still hypotheses since they are derived by formal manipulations on divergent double series. We verify the results numerically, which require a computation of the Euler and related sums with high precision. Some old and new algorithms for this purpose are commented.

Full Text: PDF     DOI: 10.15640/arms.v10n1a4