A New Closed-Form Formula of the Gauss Hypergeometric Function at Specific Arguments

Abstract

In this paper, the authors briefly review some closed-form formulas of the Gauss hypergeometric function at specific arguments, alternatively prove four of these formulas, newly extend a closed-form formula of the Gauss hypergeometric function at some specific arguments, successfully apply a special case of the newly extended closed-form formula to derive an alternative form for the Maclaurin power series expansion of the Wilf function, and discover two novel increasing rational approximations to a quarter of the circular constant.

Keywords

• Gauss hypergeometric function
• Euler integral representation
• Lerch transcendent
• specific argument
• closed-form formula
• contiguous function