I series expanded the following expression in Mathematica and the result is independent of z: $$(1-z)^{-a }\left(\, _2F_1(1,-a ;1-a ;1-z)+\, _2F_1\left(1,a ;a+1;\frac{1}{1-z}\right)-1\right)-(1-z)^{a } \left(\, _2F_1\left(1,-a ;1-a ;\frac{1}{1-z}\right)+\, _2F_1(1,a ;a +1;1-z)-1\right)$$
I was trying to find some relations of the hypergeometric function to prove this, but I cannot find any. Anyone know what relation of the hypergeometric function is useful in proving this?