# §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

 31.15.1 $\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+\left(\sum_{j=1}^{N}\frac{\gamma_{% j}}{z-a_{j}}\right)\frac{\mathrm{d}w}{\mathrm{d}z}+\frac{\Phi(z)}{\prod_{j=1}^% {N}(z-a_{j})}w=0,$

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

## §31.15(ii) Zeros

If $z_{1},z_{2},\dots,z_{n}$ 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

 31.15.2 $\sum_{j=1}^{N}\frac{\gamma_{j}/2}{z_{k}-a_{j}}+\sum_{\begin{subarray}{c}j=1\\ j\neq k\end{subarray}}^{n}\frac{1}{z_{k}-z_{j}}=0,$ $k=1,2,\dots,n$.

If $t_{k}$ is a zero of the Van Vleck polynomial $V(z)$, corresponding to an $n$th degree Stieltjes polynomial $S(z)$, and $z_{1}^{\prime},z_{2}^{\prime},\dots,z_{n-1}^{\prime}$ 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

 31.15.3 $\sum_{j=1}^{N}\frac{\gamma_{j}}{t_{k}-a_{j}}+\sum_{j=1}^{n-1}\frac{1}{t_{k}-z_% {j}^{\prime}}=0.$

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 $z_{k}$, $k=1,2,\ldots,n$, of the Stieltjes polynomial $S(z)$ are the critical points of the function $G$, that is, points at which $\ifrac{\partial G}{\partial\zeta_{k}=0}$, $k=1,2,\ldots,n$, where

 31.15.4 $G(\zeta_{1},\zeta_{2},\dots,\zeta_{n})=\prod_{k=1}^{n}\prod_{\ell=1}^{N}(\zeta% _{k}-a_{\ell})^{\ifrac{\gamma_{\ell}}{2}}\prod_{j=k+1}^{n}(\zeta_{k}-\zeta_{j}).$

If the following conditions are satisfied:

 31.15.5 $\displaystyle\gamma_{j}$ $\displaystyle>0,$ $\displaystyle a_{j}$ $\displaystyle\in\mathbb{R}$, $j=1,2,\dots,N$,

and

 31.15.6 $a_{j} $j=1,2,\dots,N-1$, ⓘ Symbols: $j$: nonnegative integer, $a$: complex parameter and $N+1$: number of singularities Referenced by: §31.15(iii) Permalink: http://dlmf.nist.gov/31.15.E6 Encodings: TeX, pMML, png See also: Annotations for §31.15(ii), §31.15 and Ch.31

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

 31.15.7 $q_{j}=\gamma_{j}\sum_{k=1}^{n}\frac{1}{z_{k}-a_{j}},$ $j=1,2,\dots,N$.

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 $\mathbf{m}=(m_{1},m_{2},\dots,m_{N-1})$, 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 $j=1,2,\dots,N-1$. We denote this Stieltjes polynomial by $S_{\mathbf{m}}(z)$.

Let $S_{\mathbf{m}}(z)$ and $S_{\mathbf{l}}(z)$ be Stieltjes polynomials corresponding to two distinct multi-indices $\mathbf{m}=(m_{1},m_{2},\dots,m_{N-1})$ and $\mathbf{l}=(\ell_{1},\ell_{2},\dots,\ell_{N-1})$. The products

 31.15.8 $S_{\mathbf{m}}(z_{1})S_{\mathbf{m}}(z_{2})\cdots S_{\mathbf{m}}(z_{N-1}),$ $z_{j}\in(a_{j},a_{j+1})$,
 31.15.9 $S_{\mathbf{l}}(z_{1})S_{\mathbf{l}}(z_{2})\cdots S_{\mathbf{l}}(z_{N-1}),$ $z_{j}\in(a_{j},a_{j+1})$,

are mutually orthogonal over the set $Q$:

 31.15.10 $Q=(a_{1},a_{2})\times(a_{2},a_{3})\times\cdots\times(a_{N-1},a_{N}),$ ⓘ Defines: $Q$: set (locally) Symbols: $(\NVar{a},\NVar{b})$: open interval, $a$: complex parameter and $N+1$: number of singularities Permalink: http://dlmf.nist.gov/31.15.E10 Encodings: TeX, pMML, png See also: Annotations for §31.15(iii), §31.15 and Ch.31

with respect to the inner product

 31.15.11 $(f,g)_{\rho}=\int_{Q}f(z)\overline{g(z)}\rho(z)\mathrm{d}z,$

with weight function

 31.15.12 $\rho(z)=\left(\prod_{j=1}^{N-1}\prod_{k=1}^{N}|z_{j}-a_{k}|^{\gamma_{k}-1}% \right)\left(\prod_{j ⓘ Defines: $\rho(z)$: weight function (locally) Symbols: $z$: complex variable, $\gamma$: real or complex parameter, $j$: nonnegative integer, $a$: complex parameter and $N+1$: number of singularities Permalink: http://dlmf.nist.gov/31.15.E12 Encodings: TeX, pMML, png See also: Annotations for §31.15(iii), §31.15 and Ch.31

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).