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In this paper the index transformation $$F\left(\tau \right)={\int }_0^{\infty }f\left(t\right){}_2{F}_1(\mu +\frac{1}{2}+i\tau ,\mu +\frac{1}{2}-i\tau ;\mu +1;-t)t^{\alpha }dt$$ ${}_2{F}_1(\mu +\frac{1}{2}+i\tau ,\mu +\frac{1}{2}-i\tau ;\mu +1;-t)$ being the Gauss hypergeometric function, is defined on certain space of generalized functions and its inversion formula established for distributions of compact support on $𝐈=(0,\infty )$.