= Euler's hypergeometric transformation
{c}
{title2=$F(a,b;c;z)=(1-z)^{c-a-b}F(c-a,c-b;c;z)$}
= Euler hypergeometric transformation
{c}
{synonym}
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.
Back to article page
