A generalization of the Heine--Stieltjes theorem

Details A generalization of the Heine--Stieltjes theorem A generalization of the Heine--Stieltjes theorem

2009-07-03

arxiv.org/abs/0907.0648v3

Constr. Approx. 34 (2011), no. 1, 135-148

Mathematics

Classical Analysis and ODEs

12 pages, typos corrected and refined the interlacing theorems

Scientific paper

10.1007/s00365-010-9102-y

We extend the Heine-Stieltjes Theorem to concern all (non-degenerate) differential operators preserving the property of having only real zeros. This solves a conjecture of B. Shapiro. The new methods developed are used to describe intricate interlacing relations between the zeros of different pairs of solutions. This extends recent results of Bourget, McMillen and Vargas for the Heun equation and answers their question on how to generalize their results to higher degrees. Many of the results are new even for the classical case.

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