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In the work, the author derives an integral representation of the Gauss hypergeometric functionsF12⁢(a-12,a;a+12;z){{}_{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.

An integral representation of the Gauss hypergeometric functions and its applications