Zeros of Jacobi functions of second kind

The number of zeros in (-1,1) of the Jacobi function of second kind Qn(α, β) (x), α, 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, is determined for every n ∈ N and for all values of the parameters α > -1 and β > -1. It turns out that this number depends essentially on α and β 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.