Zeros of Jacobi functions of second kind

The number of zeros in ( - 1 , 1 ) of the Jacobi function of second kind Q n ( α , β ) ( x ) , α , β > - 1 , i.e. the second solution of the differential equation ( 1 - x 2 ) 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 Q n ( 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 Laguerreʹs and Hermiteʹs functions of second kind.