Global Sturm inequalities for the real zeros of the solutions of the Gauss hypergeometric differential equation

Abstract

Liouville-Green transformations of the Gauss hypergeometric equation with changes of variable $$z(x)=\int\sp xt\sp {p-1}(1-t)\sp {q-1}dt$$ are considered. When $p+q=1,\ p=0$ or $q=0$ these transformations, together with the application of Sturm theorems, lead to properties satisfied by all the real zeros $x_i$ of any of its solutions in the interval $(0,1)$. Global bounds on the differences $z(x_{k+1})-z(x_k),$ with $ 0<x_k<x_{k+1}<1$ being consecutive zeros, and monotonicity of their distances as a function of $k$ can be obtained. We investigate the parameter ranges for which these two different Sturm-type properties are available. Classical results for Jacobi polynomials (Szegő's bounds, Grosjean's inequality) are particular cases of these more general properties. Similar properties are found for other values of $p$ and $q$, particularly when $ alpha and $ beta , where $\alpha$ and $\beta$ are the usual Jacobi parameters.