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

Liouville-Green transformations of the Gauss hypergeometric equation with changes of variable z(x) = ∫ x tp−1(1 − t)q−1dt 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 xi of any of its solutions in the interval (0, 1). Global bounds on the differences z(xk+1) − z(xk), 0 < xk < xk+1 < 1 being consecutive zeros, and monotonicity of these 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 |p| = |α| and |q| = |β|, α and β being the usual Jacobi parameters.