# §31.15 Stieltjes Polynomials

## §31.15(i) Definitions

Stieltjes polynomials are polynomial solutions of the Fuchsian equation (31.14.1). Rewrite (31.14.1) in the form

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.

## §31.15(ii) Zeros

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:

and

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

See Marden (1966), Alam (1979), and Al-Rashed and Zaheer (1985) for further results on the location of the zeros of Stieltjes and Van Vleck polynomials.

## §31.15(iii) Products of Stieltjes Polynomials

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

The normalized system of products (31.15.8) forms an orthonormal basis in the Hilbert space . For further details and for the expansions of analytic functions in this basis see Volkmer (1999).