Heine–Stieltjes polynomials

In mathematics, the Heine–Stieltjes polynomials or Stieltjes polynomials, introduced by T. J. Stieltjes (1885), are polynomial solutions of a second-order Fuchsian equation, a differential equation all of whose singularities are regular. The Fuchsian equation has the form

[math]\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 }[/math]

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

• Marden, Morris (1931), "On Stieltjes Polynomials", Transactions of the American Mathematical Society (Providence, R.I.: American Mathematical Society) 33 (4): 934–944, doi:10.2307/1989516, ISSN 0002-9947 
• Sleeman, B. D.; Kuznetzov, V. B. (2010), "Stieltjes Polynomials", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F. et al., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, http://dlmf.nist.gov/31.15 
• Stieltjes, T. J. (1885), "Sur certains polynômes qui vérifient une équation différentielle linéaire du second ordre et sur la theorie des fonctions de Lamé", Acta Mathematica 6 (1): 321–326, doi:10.1007/BF02400421