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§31.15 Stieltjes Polynomials

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§31.15(i) Definitions

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

31.15.1 2wz2+(j=1Nγjz-aj)wz+Φ(z)j=1N(z-aj)w=0,

where Φ(z) is a polynomial of degree not exceeding N-2. There exist at most (n+N-2N-2) polynomials V(z) of degree not exceeding N-2 such that for Φ(z)=V(z), (31.15.1) has a polynomial solution w=S(z) of degree n. The V(z) are called Van Vleck polynomials and the corresponding S(z) Stieltjes polynomials.

§31.15(ii) Zeros

If z1,z2,,zn are the zeros of an nth degree Stieltjes polynomial S(z), then every zero zk is either one of the parameters aj or a solution of the system of equations

31.15.2 j=1Nγj/2zk-aj+j=1jkn1zk-zj=0,
k=1,2,,n.

If tk is a zero of the Van Vleck polynomial V(z), corresponding to an nth degree Stieltjes polynomial S(z), and z1,z2,,zn-1 are the zeros of S(z) (the derivative of S(z)), then tk is either a zero of S(z) or a solution of the equation

31.15.3 j=1Nγjtk-aj+j=1n-11tk-zj=0.

The system (31.15.2) determines the zk as the points of equilibrium of n movable (interacting) particles with unit charges in a field of N particles with the charges γj/2 fixed at aj. This is the Stieltjes electrostatic interpretation.

The zeros zk, k=1,2,,n, of the Stieltjes polynomial S(z) are the critical points of the function G, that is, points at which G/ζk=0, k=1,2,,n, where

31.15.4 G(ζ1,ζ2,,ζn)=k=1n=1N(ζk-a)γ/2j=k+1n(ζk-ζj).

If the following conditions are satisfied:

31.15.5 γj >0,
aj ,
j=1,2,,N,

and

31.15.6 aj<aj+1,
j=1,2,,N-1,

then there are exactly (n+N-2N-2) polynomials S(z), each of which corresponds to each of the (n+N-2N-2) ways of distributing its n zeros among N-1 intervals (aj,aj+1), j=1,2,,N-1. In this case the accessory parameters qj are given by

31.15.7 qj=γjk=1n1zk-aj,
j=1,2,,N.

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 m=(m1,m2,,mN-1), where each mj is a nonnegative integer, there is a unique Stieltjes polynomial with mj zeros in the open interval (aj,aj+1) for each j=1,2,,N-1. We denote this Stieltjes polynomial by Sm(z).

Let Sm(z) and Sl(z) be Stieltjes polynomials corresponding to two distinct multi-indices m=(m1,m2,,mN-1) and l=(1,2,,N-1). The products

31.15.8 Sm(z1)Sm(z2)Sm(zN-1),
zj(aj,aj+1),
31.15.9 Sl(z1)Sl(z2)Sl(zN-1),
zj(aj,aj+1),

are mutually orthogonal over the set Q:

31.15.10 Q=(a1,a2)×(a2,a3)××(aN-1,aN),

with respect to the inner product

31.15.11 (f,g)ρ=Qf(z)g¯(z)ρ(z)z,

with weight function

31.15.12 ρ(z)=(j=1N-1k=1N|zj-ak|γk-1)(j<kN-1(zk-zj)).

The normalized system of products (31.15.8) forms an orthonormal basis in the Hilbert space Lρ2(Q). For further details and for the expansions of analytic functions in this basis see Volkmer (1999).