Polynomial Solutions of the Heun Equation

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.


Introduction
We study polynomial solutions of the Heun equation where Q, P , and V are given polynomials.Q is a polynomial of degree k, P is at most of degree k − 1, and V is at most of degree k − 2. E. Heine and T. Stieltjes posed the following problem: Problem.Given a pair of polynomials {Q, P } and a positive integer n find all polynomials V such that (1) has a polynomial solution S of degree n.Polynomials V are referred to as Van Vleck polynomials and polynomials S as Stieltjes polynomials.
For a generic pair {Q, P } there exist n+k−2 n distinct Van Vleck polynomials.
The simplest case is k = 2, when equation ( 1) is an equation of hypergeometric type: Q is quadratic, P is at most linear and V reduces to a (spectral) parameter.This situation was thoroughly studied in the past and all polynomial solutions are brought to six types of either finite or infinite systems of orthogonal polynomials e.g.[4].Asymptotic distribution of zeros of orthogonal polynomials has been studied for quite a long time and many important results are known [13].
Even this problem has a long history, going back to G. Lamé.Already Heine and Stieltjes knew that for a fixed n the above mentioned problem has n + 1 solutions, i.e. that there exist n + 1 distinct Van Vleck polynomials.Moreover, in the case of the Lamé equation (P = Q /2) and if we additionally assume that Q has three real and distinct roots a 1 < a 2 < a 3 then each root of each V and each S is real and simple, the roots of V and S lie between a 1 and a 3 , none of the roots of S coincides with any a i (i = 1, 2, 3), and n + 1 polynomials S can be distinguished by the number of roots lying in the interval (a 1 , a 2 ) (the remaining roots lie in (a 2 , a 3 )) [14].Besides this, there is no zero of S between a 2 and the zero of the corresponding Van Vleck polynomial [1], cf. Figure 1.
Some additional results are known for fixed n.Each Van Vleck (linear) polynomial has a single zero ν i , i = 1, . . ., n + 1.We can form a so-called spectral polynomial made of these zeros Zeros of two successive spectral polynomials, i.e.Sp n and Sp n+1 interlace: between any two roots of Sp n lies a root of Sp n+1 , and vice versa [2].On the other hand, in spite of the fact that these polynomials have simple zeros that interlace, the system {Sp n } ∞ n=1 is not orthogonal with respect to any measure.The proof in [2] is based on the finding that the asymptotic zero distribution of Sp n [3] is different from that of orthogonal polynomials, showing also that Sp n do not obey any three-term recurrence relation.
As already mentioned above, the roots of Van Vleck's ν i lie between a 1 and a 3 , and are mutually different, making it thus possible to order Stieltjes polynomials accordingly.So, for a fixed n, we have a sequence of n + 1 Stieltjes polynomials S (n) i of degree n, i = 1, . . ., n+1.Two interesting results are proved in [1].The n zeros of and S (n+1) j interlace if and only if i = j or i = j + 1, otherwise they do not interlace.There is no definitive answer to the question of orthogonality of S (n) i .If complex roots of Q are admitted, G. Pólya proved [9] that all roots of both V and S belong to the convex hull Conv Q of a 1 , a 2 , a 3 provided that all residues of P/Q are positive.
Investigations of the root asymptotics of both Van Vleck and Stieltjes polynomials have a considerably shorter history.We summarize here some salient results [10][11][12].The roots can be asymptotically localized.For any > 0 there exist N such that for any n ≥ N any root of any V as well as any root of the corresponding S lie in the -neighbourhood (in the usual Euclidean distance on C) of the convex hull of a 1 , a 2 , a 3 .This result shows that the asymptotic behaviour of roots is determined by Q, i.e. it is not influenced by P for sufficiently large n.
For a more detailed description of asymptotic distribution we associate to each polynomial p n a finite real measure where δ(z − z j ) is the Dirac measure supported at the root z j .This probability measure is referred to as the root-counting measure of the polynomial p n .Now, two questions are to be answered.Does the sequence {μ n } converge (in the weak sense) to a limiting measure μ and if so what does μ look like?We may ask these questions when p n = Sp n .The first question is answered positively [11,12].The sequence {μ n } of the root-counting measures of its spectral polynomials converges to a probability measure μ supported on the union of three curves located inside Conv Q and connecting the three roots of Q with a certain interior point, cf.The support of μ is a union of three curve segments γ i , i ∈ {1, 2, 3}.They may be described as the set of all b ∈ Conv Q satisfying here j and k are the remaining two indices in {1, 2, 3} in any order and the integration is taken over the straight interval connecting a j and a k .We can see that a i belong to γ i and that these three curves connect the corresponding a i with a common point within Conv Q .Take a segment of γ i connecting a i with the common intersection point of all γ's.Let us denote the union of these three segments by Γ Q .
Then the support of the limiting root-counting measure μ coincides with Γ Q .Knowing the support of μ it is also possible to define its density along the support using the linear differential equation satisfied by its Cauchy transform [11] In the case when Q(z) has all real zeros, the density is explicitly given in [3].The Cauchy transform C ν (z) and the logarithmic potential pot ν (z) of a (complex-valued) measure ν supported in C are given by: C ν (z) is analytic outside the support of ν [5].
In [11] we were able to find an additional probability measure ν which is easily described and from which the measure μ is obtained by the inverse balayage, i.e. the support of μ will be contained in the support of the measure ν and they have the same logarithmic potential outside the support of the latter one.This measure is uniquely determined by the choice of a root of Q(z), and thus we in fact have constructed three different measures ν i having the same measure μ as their inverse balayage.
Let us try to formulate similar results for the asymptotic root behaviour of Stieltjes polynomials.
To this end we must formulate in more detail which sequence of polynomials we are studying.Take a sequence of monic (the leading coefficient is 1) Van Vleck polynomials { V n } converging to some monic linear polynomial V .The existence of a linear polynomial V is ensured by the existence of the limit of the sequence of (unique) roots ν n,in of { V n }.The above mentioned results guarantee the existence of plenty of such converging sequences in Conv Q and the limit ν of these roots must necessarily belong to Γ Q .
Having chosen { V n } we take any sequence of the corresponding {S n,in }, deg S n,in = n whose corresponding sequence { V n } has a limit.If we denote by μ n,in the root-counting measure of the corresponding Stieltjes polynomial, we have proved that the sequence {μ n,in } converges weakly to the unique probability measure μ V whose Cauchy transform C V (z) satisfies the equation almost everywhere in C.
In order to formulate further results we used [12] the notion of the quadratic differential (cf.also [7,8]).We avoid this way of formulating the results, because it would necessarily exceed the scope if this paper.Instead, we limit ourselves to presenting a typical example, cf. the right part of Figure 2. The support of the limit measure consists of singular trajectories of the quadratic differential.They run close to the roots shown in red.In this particular case, one trajectory joins two zeros of Q and the other one joins the third zero of Q with the root of the limiting Van Vleck polynomial.

Bispectral problems
Concerning the situation when k = 4 certain general statements have already been published (e.g. in [6,7]).In the case when the roots of Van Vleck and Stieltjes polynomials are real we can still rely on the result of Stieltjes mentioned above, which make ordering of Stieltjes polynomials possible.The situation is shown in Figure 3.
When complex roots come into play, the picture is less clear.Figure 3 suggests that the asymptotic root distribution of Van Vleck polynomials has a more complicated structure than before.On the other hand, the structure of the asymptotic root distribution of Stieltjes polynomials bears some resemblance to the k = 3 case.
There are still several questions open.In addition, many other unsolved problems can be found for higher linear differential equations with polynomial coefficients.

Fig. 1 :Fig. 2 :
Fig.1: The situation for P = 0 and n = 25.The thick black dots mark the roots of Q(x) = (x + 2)(x − 1)(x − 4), the thick green dots mark the roots of n + 1 Van Vleck polynomials, and the small red dots mark n roots of the corresponding Stieltjes polynomials