Euler's hypergeometric transformation ($F(a,b;c;z)=(1-z{)}^{c-a-b}F(c-a,c-b;c;z)$)

Multiply a solution with parameters $(c-a,c-b;c)$ by $(1-z{)}^{c-a-b}$. Its local exponents become exactly those of the Gauss hypergeometric equation with parameters $(a,b;c)$: those at one shift by $c-a-b$, and those at infinity shift by its negative. Direct differentiation gives the same differential equation. Near zero, choose the factor with value one; uniqueness of the analytic normalized solution, for $c$ not a nonpositive integer, gives the identity. Compatible analytic continuation extends it to other domains and parameters where the normalized functions are defined.