Zeros of Jacobi functions of second kind

The number of zeros in (-1,1)(-1,1) of the Jacobi function of second kind Qn(α,β)(x), α,β>-1α,β>-1, i.e. the second solution of the differential equation(1-x2)y″(x)+(β-α-(α+β+2)x)y′(x)+n(n+α+β+1)y(x)=0,(1-x2)y″(x)+(β-α-(α+β+2)x)y′(x)+n(n+α+β+1)y(x)=0,is determined for every n∈Nn∈N and for all values of the parameters α>-1α>-1 and β>-1β>-1. It turns out that this number depends essentially on αα and ββ as well as on the specific normalization of the function Qn(α,β)(x). Interlacing properties of the zeros are also obtained. As a consequence of the main result, we determine the number of zeros of Laguerre's and Hermite's functions of second kind.