31 Heun Functions Properties 31.14 General Fuchsian Equation 31.16 Mathematical Applications

§31.15 Stieltjes Polynomials

Referenced by:: §31.5
Permalink:: http://dlmf.nist.gov/31.15

§31.15(i) Definitions
§31.15(ii) Zeros
§31.15(iii) Products of Stieltjes Polynomials

§31.15(i) Definitions

Notes:: See Marden (1966).
Keywords:: Fuchsian equation, Stieltjes polynomials, Van Vleck polynomials
Permalink:: http://dlmf.nist.gov/31.15.i

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

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

Symbols:: $\frac{df}{dx}$ : derivative of with respect to , : complex variable, $\gamma$ : real or complex parameter, : nonnegative integer, : complex parameter, : number of singularities and $\Phi(z)$ : polynomial
Referenced by:: §31.15(i)
Permalink:: http://dlmf.nist.gov/31.15.E1
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where $\Phi(z)$ is a polynomial of degree not exceeding . There exist at most $\binom{n+N-2}{N-2}$ polynomials of degree not exceeding such that for $\Phi(z)=V(z)$ , (31.15.1) has a polynomial solution of degree . The are called Van Vleck polynomials and the corresponding Stieltjes polynomials.

§31.15(ii) Zeros

Keywords:: Stieltjes polynomials, Van Vleck polynomials, electric particle field
Permalink:: http://dlmf.nist.gov/31.15.ii

If $z_{1},z_{2},\dots,z_{n}$ are the zeros of an th degree Stieltjes polynomial , then every zero $z_{k}$ is either one of the parameters $a_{j}$ or a solution of the system of equations

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

Symbols:: : complex variable, $\gamma$ : real or complex parameter, : nonnegative integer, : nonnegative integer, : complex parameter and : number of singularities
Referenced by:: §31.15(ii)
Permalink:: http://dlmf.nist.gov/31.15.E2
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If $t_{k}$ is a zero of the Van Vleck polynomial , corresponding to an th degree Stieltjes polynomial , and $z_{1}^{{\prime}},z_{2}^{{\prime}},\dots,z_{{n-1}}^{{\prime}}$ are the zeros of $S^{{\prime}}(z)$ (the derivative of ), 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:: : complex variable, $\gamma$ : real or complex parameter, : nonnegative integer, : nonnegative integer, : complex parameter, : number of singularities and $t_{k}$ : zero of
Permalink:: http://dlmf.nist.gov/31.15.E3
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The system (31.15.2) determines the $z_{k}$ as the points of equilibrium of movable (interacting) particles with unit charges in a field of 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 are the critical points of the function , that is, points at which $\ifrac{\partial G}{\partial\zeta_{k}=0}$ , $k=1,2,\ldots,n$ , where

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

Symbols:: $\gamma$ : real or complex parameter, : nonnegative integer, : nonnegative integer, : complex parameter and : number of singularities
Permalink:: http://dlmf.nist.gov/31.15.E4
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If the following conditions are satisfied:

31.15.5

$\gamma_{j}>0,$

$a_{j}\in\Real$ , $j=1,2,\dots,N$ ,

Symbols:: $\in$ : element of, $\Real$ : real line (excluding infinity), $\gamma$ : real or complex parameter, : nonnegative integer, : complex parameter and : number of singularities
Referenced by:: §31.15(iii)
Permalink:: http://dlmf.nist.gov/31.15.E5
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and

31.15.6 $a_{j}<a_{{j+1}},$ $j=1,2,\dots,N-1$ ,

Symbols:: : nonnegative integer, : complex parameter and : number of singularities
Referenced by:: §31.15(iii)
Permalink:: http://dlmf.nist.gov/31.15.E6
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then there are exactly $\binom{n+N-2}{N-2}$ polynomials , each of which corresponds to each of the $\binom{n+N-2}{N-2}$ ways of distributing its zeros among 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 $q_{j}=\gamma_{j}\sum_{{k=1}}^{n}\frac{1}{z_{k}-a_{j}},$ $j=1,2,\dots,N$ .

Symbols:: : complex variable, $\gamma$ : real or complex parameter, : nonnegative integer, : nonnegative integer, : complex parameter, : real or complex parameter and : number of singularities
Permalink:: http://dlmf.nist.gov/31.15.E7
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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

Keywords:: Hilbert space, Stieltjes polynomials
Permalink:: http://dlmf.nist.gov/31.15.iii

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

Symbols:: $\in$ : element of, : open interval, : complex variable, : nonnegative integer, : complex parameter, : number of singularities and $S_{{\mathbf{m}}}(z)$ : Stieltjes polynomial
Referenced by:: §31.15(iii)
Permalink:: http://dlmf.nist.gov/31.15.E8
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31.15.9 $S_{{\mathbf{l}}}(z_{1})S_{{\mathbf{l}}}(z_{2})\cdots S_{{\mathbf{l}}}(z_{{N-1}% }),$ $z_{j}\in(a_{j},a_{{j+1}})$ ,

Symbols:: $\in$ : element of, : open interval, : complex variable, : nonnegative integer, : complex parameter, : number of singularities and $S_{{\mathbf{m}}}(z)$ : Stieltjes polynomial
Permalink:: http://dlmf.nist.gov/31.15.E9
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are mutually orthogonal over the set :

31.15.10 $Q=(a_{1},a_{2})\times(a_{2},a_{3})\times\cdots\times(a_{{N-1}},a_{{N}}),$

Symbols:: : open interval, : complex parameter, : number of singularities and : set
Permalink:: http://dlmf.nist.gov/31.15.E10
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with respect to the inner product

31.15.11 $(f,g)_{\rho}=\int_{Q}f(z)\bar{g}(z)\rho(z)dz,$

Symbols:: : differential of , $\int$ : integral, : open interval, : complex variable, : set and $\rho(z)$ : weight function
Permalink:: http://dlmf.nist.gov/31.15.E11
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with weight function

31.15.12 $\rho(z)=\left(\prod_{{j=1}}^{{N-1}}\prod_{{k=1}}^{N}|z_{j}-a_{k}|^{{\gamma_{k}% -1}}\right)\left(\prod_{{j<k}}^{{N-1}}(z_{k}-z_{j})\right).$

Symbols:: : complex variable, $\gamma$ : real or complex parameter, : nonnegative integer, : complex parameter, : number of singularities and $\rho(z)$ : weight function
Permalink:: http://dlmf.nist.gov/31.15.E12
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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).

© 2010–2013 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.6; Release date 2013-05-06 Math-Image version (See MathML) . A Printed Companionis available. 31.14 General Fuchsian Equation 31.16 Mathematical Applications

§31.15 Stieltjes Polynomials

Contents

§31.15(i) Definitions

§31.15(ii) Zeros

§31.15(iii) Products of Stieltjes Polynomials