In mathematics, the Heine–Stieltjes polynomials or Stieltjes polynomials, introduced by , are polynomial solutions of a second-order
Fuchsian equation
In mathematics, in the theory of ordinary differential equations in the complex plane \Complex, the points of \Complex are classified into ''ordinary points'', at which the equation's coefficients are analytic functions, and ''singular points'', at ...
, a
differential equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
all of whose singularities are
regular. The Fuchsian equation has the form
:
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 polynomial
In mathematics, the local Heun function H \ell (a,q;\alpha ,\beta, \gamma, \delta ; z) is the solution of Heun's differential equation that is holomorphic and 1 at the singular point ''z'' = 0. The local Heun function is called a Heun ...
s 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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