The classical Heun equation has the form

Dedicated to Professor Masaki Kashiwara on his sixtieth birthday Abstract. We obtain integral representations of solutions to special cases of the Fuchsian system of differential equations and Heun’s differential equation. In particular, we calculate the monodromy of solutions to the Fuchsian equation that corresponds to Picard’s solution of the sixth Painlevé equation, and to Heun’s equation. 1.

Abstract. We establish a hierarchy of weighted majorization relations for the singularities of generalized Lamé equations and the zeros of their Van Vleck and Heine-Stieltjes polynomials as well as for multiparameter spectral polynomials of higher Lamé operators. These relations translate into natural dilation and subordination properties in the Choquet order for certain probability measures associated with the aforementioned polynomials. As a consequence we obtain new inequalities for the moments and logarithmic potentials of the corresponding root-counting measures and their weak- ∗ limits in the semi-classical and various thermodynamic asymptotic regimes. We also prove analogous results for systems of orthogonal polynomials such as Jacobi polynomials. 1.

Abstract. We review several results on the finite-gap potential and Heun’s differential equation, and we discuss relationships among the finite-gap potential, the WKB analysis and Heun’s differential equation. 1.

Abstract. The well-known Heun equation has the form j Q(z) d2 ff d + P(z) + V (z) S(z) = 0, dz2 dz where Q(z) is a cubic complex polynomial, P(z) and V (z) are polynomials of degree at most 2 and 1 respectively. One of the classical problems about the Heun equation suggested by E. Heine and T. Stieltjes in the late 19-th century is for a given positive integer n to find all possible polynomials V (z) such that the above equation has a polynomial solution S(z) of degree n. Below we prove a conjecture of the second author, see [17] claiming that the union of the roots of such V (z)’s for a given n tends when n → ∞ to a certain compact connecting the three roots of Q(z) which is given by a condition that a certain

Abstract. The well-known Heun equation has the form Q(z) d2 d + P (z) + V (z) S(z) = 0, dz2 dz where Q(z) is a cubic complex polynomial, P (z) and V (z) are polynomials of degree at most 2 and 1 respectively. One of the classical problems about the Heun equation suggested by E. Heine and T. Stieltjes in the late 19-th century is for a given positive integer n to find all possible polynomials V (z) such that the above equation has a polynomial solution S(z) of degree n. Below we prove a conjecture of the second author, see [17] claiming that the union of the roots of such V (z)’s for a given n tends when n → ∞ to a certain compact connecting the three roots of Q(z) which is given by a condition that a certain natural abelian integral is real-valued, see Theorem 2. In particular, we prove several new results of independent interest about rational Strebel differentials.

We review properties of certain types of polynomial solutions of the Heun equation. Two aspects are particularly concerned, the interlacing property of spectral and Stieltjes polynomials in the case of real roots of these polynomials and asymptotic root distribution when complex roots are present. 1

Polynomial solutions to the generalized Lamé equation, the Stieltjes polynomials, and the associated Van Vleck polynomials have been studied since the 1830’s, beginning with Lamé in his studies of the Laplace equation on an ellipsoid, and in an ever widening variety of applications since. In this paper we show how the zeros of Stieltjes polynomials are distributed and present two new interlacing theorems. We arrange the Stieltjes polynomials according to their Van Vleck zeros and show, firstly, that the zeros of successive Stieltjes polynomials of the same degree interlace, and secondly, that the zeros of Stieltjes polynomials of successive degrees interlace. We use these results to deduce new asymptotic properties of Stieltjes and Van Vleck polynomials. We also show that no sequence of Stieltjes polynomials is orthogonal.

Abstract. The classical Heun equation has the form j Q(z) d2 ff d + P(z) + V (z) S(z) = 0, dz2 dz where Q(z) is a cubic complex polynomial, P(z) is a polynomial of degree at most 2 and V (z) is at most linear. In the second half of the nineteenth century E. Heine and T. Stieltjes in [5], [13] initiated the study of the set of all V (z) for which 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 → ∞. We formulate an intriguing conjecture of K. Takemura describing the limiting set and give a substantial amount of additional information obtained using some technique developed in [7].