Heine–Stieltjes polynomials explained

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

(d ^{2} S)/(dz ^{2}) + \left(\sum ^{N} ^{j =1} (\gamma _{j})/(z - a _{j}) \right) (dS)/(dz) + (V(z))/(\prod ^{N} ^{j =1} (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.