An integral representation of the Gauss hypergeometric functions and its applications

Abstract In the work, the author derives an integral representation of the Gauss hypergeometric functions <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mmultiscripts> <m:mi>F</m:mi> <m:mn>1</m:mn> <m:none/> <m:mprescripts/> <m:mn>2</m:mn> <m:none/> </m:mmultiscripts> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>a</m:mi> <m:mo>-</m:mo> <m:mfrac> <m:mn>1</m:mn> <m:mn>2</m:mn> </m:mfrac> </m:mrow> <m:mo>,</m:mo> <m:mi>a</m:mi> <m:mo>;</m:mo> <m:mrow> <m:mi>a</m:mi> <m:mo>+</m:mo> <m:mfrac> <m:mn>1</m:mn> <m:mn>2</m:mn> </m:mfrac> </m:mrow> <m:mo>;</m:mo> <m:mi>z</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {{}_{2}F_{1}(a-\frac{1}{2},a;a+\frac{1}{2};z)} by three approaches, applies the integral representation to give integral representations of several functions involving the inverse tangent function and including the Wilf function, and find out several combinatorial identities.