Heine–Stieltjes polynomials

Template:About Template:Format footnotes In mathematics, the Heine–Stieltjes polynomials or Stieltjes polynomials, introduced by Template:Harvs, are polynomial solutions of a second-order Fuchsian equation, a differential equation all of whose singularities are regular. The Fuchsian equation has the form

$\frac{d^{2}S}{d{z}^2}+\left(\sum _{j=1}^N\frac{{\gamma }_j}{z-a_j}\right)\frac{dS}{dz}+\frac{V(z)}{\prod _{j=1}^N(z-a_j)}S=0$

for some polynomial V(z) of degree at most N − 2, and if this has a polynomial solution S then V is called a Van Vleck polynomial (after Edward Burr Van Vleck) and S is called a Heine–Stieltjes polynomial.

Heun polynomials are the special cases of Stieltjes polynomials when the differential equation has four singular points.

References

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