Heine–Stieltjes Polynomials

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

:$\frac+\left(\sum _^N \frac \right) \frac + \fracS = 0$

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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Polynomials

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