Heine–Stieltjes polynomials

For the orthogonal polynomials, see Stieltjes-Wigert polynomial. For the polynomials associated to a family of orthogonal polynomials, see Stieltjes polynomials.

In mathematics, the Heine–Stieltjes polynomials or Stieltjes polynomials, introduced by T. J. Stieltjes,^[1] are polynomial solutions of a second-order Fuchsian equation, a differential equation all of whose singularities are regular. The Fuchsian equation has the form

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

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.