2008/12/12 by B. Shapiro, Shapiro, B., M. Tater +1
Mathematics · Physics and Astronomy · #30C15 #33E05 (Secondary) #34L20 (Primary) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #math-ph #math.CA #math.MP #msc:30C15 #msc:33E05 #msc:34L20
paper · pdf · doi:10.48550/arxiv.0812.2321
14 pages, 5 figures
arxiv created 2008/12/12 · arxiv updated 2009/12/01
The classical Heun equation has the form Q(z) d2/dz2 +P(z) d/dz +V(z)S(z)=0 where Q(z) is a cubic, P(z) at most quadratic and V(z) linear polynomials resp. In the second half of the 19-th century E.Heine and T.STieltjes initiated the study of the set of all V(z) such that the above equation has a polynomial solution S(z) of a given degree n. The main goal of the present paper is to study the union of the roots of the latter set of V(z)*s when n->oo. We formulate an intriguing conjecture of K.Takemura describing the limiting set and give a substantial amount of additional information.