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

where $\mathrm{\Phi }(z)$ is a polynomial of degree not exceeding $N-2$. There exist at most $\left(\genfrac{}{}{0.0pt}{}{n+N-2}{N-2}\right)$ polynomials $V(z)$ of degree not exceeding $N-2$ such that for $\mathrm{\Phi }(z)=V(z)$, (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.

If $z_1,z_2,\mathrm{\dots },z_n$ are the zeros of an $n$th degree Stieltjes polynomial $S(z)$, then every zero $z_k$ is either one of the parameters $a_j$ or a solution of the system of equations

If $t_k$ is a zero of the Van Vleck polynomial $V(z)$, corresponding to an $n$th degree Stieltjes polynomial $S(z)$, and $z_1^{\prime },z_2^{\prime },\mathrm{\dots },z_{n-1}^{\prime }$ are the zeros of $S^{\prime }(z)$ (the derivative of $S(z)$), then $t_k$ is either a zero of $S^{\prime }(z)$ or a solution of the equation

The system (31.15.2) determines the $z_k$ as the points of equilibrium of $n$ movable (interacting) particles with unit charges in a field of $N$ particles with the charges ${\gamma }_j/2$ fixed at $a_j$. This is the Stieltjes electrostatic interpretation.

The zeros $z_k$, $k=1,2,\mathrm{\dots },n$, of the Stieltjes polynomial $S(z)$ are the critical points of the function $G$, that is, points at which $\partial G/\partial {\zeta }_k=0$, $k=1,2,\mathrm{\dots },n$, where

If the following conditions are satisfied:

and

then there are exactly $\left(\genfrac{}{}{0.0pt}{}{n+N-2}{N-2}\right)$ polynomials $S(z)$, each of which corresponds to each of the $\left(\genfrac{}{}{0.0pt}{}{n+N-2}{N-2}\right)$ ways of distributing its $n$ zeros among $N-1$ intervals $(a_j,a_{j+1})$, $j=1,2,\mathrm{\dots },N-1$. In this case the accessory parameters $q_j$ 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.

If the exponent and singularity parameters satisfy (31.15.5)–(31.15.6), then for every multi-index $𝐦=(m_1,m_2,\mathrm{\dots },m_{N-1})$, where each $m_j$ is a nonnegative integer, there is a unique Stieltjes polynomial with $m_j$ zeros in the open interval $(a_j,a_{j+1})$ for each $j=1,2,\mathrm{\dots },N-1$. We denote this Stieltjes polynomial by $S_{𝐦}(z)$.

Let $S_{𝐦}(z)$ and $S_{𝐥}(z)$ be Stieltjes polynomials corresponding to two distinct multi-indices $𝐦=(m_1,m_2,\mathrm{\dots },m_{N-1})$ and $𝐥=({\mathrm{\ell }}_1,{\mathrm{\ell }}_2,\mathrm{\dots },{\mathrm{\ell }}_{N-1})$. The products

are mutually orthogonal over the set $Q$:

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 $L_{\rho }^2(Q)$. For further details and for the expansions of analytic functions in this basis see Volkmer (1999).