A Functional Expansion and a New Set of Rapidly Convergent Series Involving Zeta Values

摘要

We derive a power series expansion for the function x → x log (tan x) - x log x + x - Ti_2(tan x), and then use it to obtain the expansion of the function x → Ti_2(tan x); as usual, Ti_2 denotes the inverse tangent integral. Two similar expansions involving Legendre's chi-function follow from the above. These representations are used to obtain new rapidly convergent numerical series involving zeta values.