Higher-order derivatives of two arctangent-based functions and their relation to Gauss hypergeometric functions

Source abstract

Motivated by a problem on the Maclaurin expansion of \arctan \sqrt{2e^{-z}-1}/\sqrt{2e^{-z}-1}\ , the authors derive higher-order derivatives and series expansions of arctan z/z and arctan√ z/√z , express them via the Gauss hypergeometric functions {}_{2}F_{1}\ and the Chebyshev polynomials Un(x), and determine their asymptotics as n → ∞. Several new identities follow, including a trigonometric identity and an infinite series identity from two Fourier–Legendre expansions of the elliptic integral K (x). Alternative proofs of known {}_{2}F_{1}\ closed forms are also given.