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 (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

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

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.

References

• Marden, Morris (1931), "On Stieltjes Polynomials", Transactions of the American Mathematical Society, 33 (4), Providence, R.I.: American Mathematical Society: 934–944, doi:10.2307/1989516, ISSN 0002-9947, JSTOR 1989516
• Sleeman, B. D.; Kuznetzov, V. B. (2010), "Stieltjes Polynomials", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248.
• 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