§31.15 Stieltjes Polynomials

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

31.15.1		$$\frac{d^{2}w}{{dz}^2}+\left(\sum _{j=1}^N\frac{{\gamma }_j}{z-a_j}\right)\frac{dw}{dz}+\frac{\mathrm{\Phi }(z)}{{\prod }_{j=1}^N(z-a_j)}w=0,$$
Symbols: $\frac{df}{dx}$: derivative of $f$ with respect to $x$, $z$: complex variable, $\gamma$: real or complex parameter, $j$: nonnegative integer, $a$: complex parameter, $N+1$: number of singularities and $\mathrm{\Phi }(z)$: polynomial Referenced by: §31.15(i) Permalink: http://dlmf.nist.gov/31.15.E1 Encodings: TeX, pMML, png See also: Annotations for 31.15(i)

where $\mathrm{\Phi }(z)$ is a polynomial of degree not exceeding $N-2$. There exist at most $\left(\genfrac{}{}{0pt}{}{n+N-2}{N-2}\right)$ 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

31.15.2		$$\sum _{j=1}^N\frac{{\gamma }_j/2}{z_k-a_j}+\sum _{\begin{array}{c}j=1\\ j\ne k\end{array}}^n\frac{1}{z_k-z_j}=0,$$
$k=1,2,\mathrm{\dots },n$.
Symbols: $z$: complex variable, $\gamma$: real or complex parameter, $j$: nonnegative integer, $n$: nonnegative integer, $a$: complex parameter and $N+1$: number of singularities Referenced by: §31.15(ii) Permalink: http://dlmf.nist.gov/31.15.E2 Encodings: TeX, pMML, png See also: Annotations for 31.15(ii)

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

31.15.3		$$\sum _{j=1}^N\frac{{\gamma }_j}{t_k-a_j}+\sum _{j=1}^{n-1}\frac{1}{t_k-z_j^{\prime }}=0.$$
Symbols: $z$: complex variable, $\gamma$: real or complex parameter, $j$: nonnegative integer, $n$: nonnegative integer, $a$: complex parameter, $N+1$: number of singularities and $t_k$: zero of $V(z)$ Permalink: http://dlmf.nist.gov/31.15.E3 Encodings: TeX, pMML, png See also: Annotations for 31.15(ii)

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

31.15.4		$$G({\zeta }_1,{\zeta }_2,\mathrm{\dots },{\zeta }_n)=\prod _{k=1}^n\prod _{\mathrm{\ell }=1}^N{({\zeta }_k-a_{\mathrm{\ell }})}^{{\gamma }_{\mathrm{\ell }}/2}\prod _{j=k+1}^n({\zeta }_k-{\zeta }_j).$$
Symbols: $\gamma$: real or complex parameter, $j$: nonnegative integer, $n$: nonnegative integer, $a$: complex parameter and $N+1$: number of singularities Permalink: http://dlmf.nist.gov/31.15.E4 Encodings: TeX, pMML, png See also: Annotations for 31.15(ii)

If the following conditions are satisfied:

31.15.5		${\gamma }_j$	$>0,$
		$a_j$	$\in ℝ$,
	$j=1,2,\mathrm{\dots },N$,
Symbols: $\in$: element of, $ℝ$: real line, $\gamma$: real or complex parameter, $j$: nonnegative integer, $a$: complex parameter and $N+1$: number of singularities Referenced by: §31.15(iii) Permalink: http://dlmf.nist.gov/31.15.E5 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for 31.15(ii)

and

31.15.6
$j=1,2,\mathrm{\dots },N-1$,
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)

then there are exactly $\left(\genfrac{}{}{0pt}{}{n+N-2}{N-2}\right)$ polynomials $S(z)$, each of which corresponds to each of the $\left(\genfrac{}{}{0pt}{}{n+N-2}{N-2}\right)$ 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

31.15.7		$$q_j={\gamma }_j\sum _{k=1}^n\frac{1}{z_k-a_j},$$
$j=1,2,\mathrm{\dots },N$.
Symbols: $z$: complex variable, $\gamma$: real or complex parameter, $j$: nonnegative integer, $n$: nonnegative integer, $a$: complex parameter, $q$: real or complex parameter and $N+1$: number of singularities Permalink: http://dlmf.nist.gov/31.15.E7 Encodings: TeX, pMML, png See also: Annotations for 31.15(ii)

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 $\mathbf{m}=(m_1,m_2,\mathrm{\dots },m_{N-1})$, 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 $S_{\mathbf{m}}(z)$.

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

31.15.8		$$S_{\mathbf{m}}(z_1)S_{\mathbf{m}}(z_2)\mathrm{\cdots }{S}_{\mathbf{m}}(z_{N-1}),$$
$z_j\in (a_j,a_{j+1})$,
Symbols: $\in$: element of, $(a,b)$: open interval, $z$: complex variable, $j$: nonnegative integer, $a$: complex parameter, $N+1$: number of singularities and $S_{\mathbf{m}}(z)$: Stieltjes polynomial Referenced by: §31.15(iii) Permalink: http://dlmf.nist.gov/31.15.E8 Encodings: TeX, pMML, png See also: Annotations for 31.15(iii)

31.15.9		$$S_{\mathbf{l}}(z_1)S_{\mathbf{l}}(z_2)\mathrm{\cdots }{S}_{\mathbf{l}}(z_{N-1}),$$
$z_j\in (a_j,a_{j+1})$,
Symbols: $\in$: element of, $(a,b)$: open interval, $z$: complex variable, $j$: nonnegative integer, $a$: complex parameter, $N+1$: number of singularities and $S_{\mathbf{m}}(z)$: Stieltjes polynomial Permalink: http://dlmf.nist.gov/31.15.E9 Encodings: TeX, pMML, png See also: Annotations for 31.15(iii)

are mutually orthogonal over the set $Q$:

31.15.10		$$Q=(a_1,a_2)\times (a_2,a_3)\times \mathrm{\cdots }\times (a_{N-1},a_N),$$
Defines: $Q$: set (locally) Symbols: $(a,b)$: 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)

with respect to the inner product

31.15.11		$${(f,g)}_{\rho }={\int }_{Q}f(z)\overline{g}(z)\rho (z)dz,$$
Symbols: $dx$: differential of $x$, $\int$: integral, $(a,b)$: open interval, $z$: complex variable, $Q$: set and $\rho (z)$: weight function Permalink: http://dlmf.nist.gov/31.15.E11 Encodings: TeX, pMML, png See also: Annotations for 31.15(iii)

with weight function

31.15.12
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)

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).