Zeros of Jacobi functions of second kind

摘要

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.