where is a polynomial of degree not exceeding . There exist at most polynomials of degree not exceeding such that for , (31.15.1) has a polynomial solution of degree . The are called Van Vleck polynomials and the corresponding Stieltjes polynomials.
If are the zeros of an th degree Stieltjes polynomial , then every zero is either one of the parameters or a solution of the system of equations
If is a zero of the Van Vleck polynomial , corresponding to an th degree Stieltjes polynomial , and are the zeros of (the derivative of ), then is either a zero of or a solution of the equation
The system (31.15.2) determines the as the points of equilibrium of movable (interacting) particles with unit charges in a field of particles with the charges fixed at . This is the Stieltjes electrostatic interpretation.
The zeros , of the Stieltjes polynomial are the critical points of the function , that is, points at which , , where
If the following conditions are satisfied:
then there are exactly polynomials , each of which corresponds to each of the ways of distributing its zeros among intervals , . In this case the accessory parameters are given by
If the exponent and singularity parameters satisfy (31.15.5)–(31.15.6), then for every multi-index , where each is a nonnegative integer, there is a unique Stieltjes polynomial with zeros in the open interval for each . We denote this Stieltjes polynomial by .
Let and be Stieltjes polynomials corresponding to two distinct multi-indices and . The products
are mutually orthogonal over the set :
with respect to the inner product
with weight function