Acta Polytechnica Vol. 51 No. 4/2011

Polynomial Solutions of the Heun Equation

B. Shapiro, M. Tater

Abstract

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.

Keywords: Heun equation, Van Vleck and Stieltjes polynomials, asymptotic root distribution, logarithmic potential.

1 Introduction
We study polynomial solutions of the Heun equation{

Q(z)
d2

dz2
+ P (z)

d
dz

+ V (z)

}
S(z) = 0, (1)

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 poly-
nomials 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) pa-
rameter. This situation was thoroughly studied in
the past and all polynomial solutions are brought to
six types of either finite or infinite systems of orthog-
onal 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].

2 k = 3 case
Next natural step is k = 3. Even this problem has a
long history, going back to G. Lamé. Already Heine
and Stieltjes knew that for a fixed n the above men-
tioned problem has n + 1 solutions, i.e. that there
exist n + 1 distinct Van Vleck polynomials. More-
over, in the case of the Lamé equation (P = Q′/2)
and if we additionally assume that Q has three real
and distinct roots a1 < a2 < a3 then each root of
each V and each S is real and simple, the roots of V
and S lie between a1 and a3, none of the roots of S
coincides with any ai (i = 1, 2, 3), and n + 1 polyno-
mials S can be distinguished by the number of roots
lying in the interval (a1, a2) (the remaining roots lie

in (a2, a3)) [14]. Besides this, there is no zero of S
between a2 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
νi, i = 1, . . . , n + 1. We can form a so-called spectral
polynomial made of these zeros

Spn(λ) =
n+1∏
i=1

(λ − νi).

Zeros of two successive spectral polynomials, i.e. Spn
and Spn+1 interlace: between any two roots of Spn
lies a root of Spn+1, and vice versa [2]. On the other
hand, in spite of the fact that these polynomials have
simple zeros that interlace, the system {Spn}∞n=1 is
not orthogonal with respect to any measure. The
proof in [2] is based on the finding that the asymp-
totic zero distribution of Spn [3] is different from that
of orthogonal polynomials, showing also that Spn do
not obey any three-term recurrence relation.

As already mentioned above, the roots of Van
Vleck’s νi lie between a1 and a3, 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 S

(n)
i and the n zeros of S

(n)
i+1

interlace. In addition, the smallest zero of S
(n)
i+1 is

smaller than the smallest zero of S
(n)
i . Besides this,

the zeros of S
(n)
i 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 or-
thogonality 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 ConvQ of a1, a2, a3 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 re-
sults [10–12].

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Acta Polytechnica Vol. 51 No. 4/2011

−2 −1 0 1 2 3 4

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 dotsmark the roots of n+1VanVleck polynomials, and the small red dotsmark n roots of the corresponding
Stieltjes polynomials

−1 0 1 2

0

0.5

1

1.5

2

2.5

3

3.5

4

−1 0 1 2

0

0.5

1

1.5

2

2.5

3

3.5

4

Fig. 2: The left part: The roots of the spectral polynomial Sp51(λ) for Q(z) = (z +1)(z − 2)(z − 2 − 4i) and P(z) =
(z+2+2i)(z −1+3i). The thick black dotsmark the roots of Q, the green dotsmark the roots ofVanVleck polynomials.
The right part: The thick green dot marks one of the 51 Van Vleck polynomials and the small red dots mark 50 roots
of the corresponding Stieltjes polynomial

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 a1, a2, a3. 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 dis-
tribution we associate to each polynomial pn a finite

real measure

μn =
1
n

n∑
j=1

δ(z − zj ),

where δ(z − zj ) is the Dirac measure supported at
the root zj . This probability measure is referred to
as the root-counting measure of the polynomial pn.

Now, two questions are to be answered. Does
the sequence {μn} converge (in the weak sense) to a

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Acta Polytechnica Vol. 51 No. 4/2011

limiting measure μ and if so what does μ look like?
We may ask these questions when pn = Spn. The
first question is answered positively [11, 12]. The
sequence {μn} of the root-counting measures of its
spectral polynomials converges to a probability mea-
sure μ supported on the union of three curves located
inside ConvQ and connecting the three roots of Q
with a certain interior point, cf. Figure 2. Moreover,
μ depends only on Q.

The support of μ is a union of three curve seg-
ments γi, i ∈ {1, 2, 3}. They may be described as the
set of all b ∈ ConvQ satisfying∫ ak

aj

√
b − t

(t − a1)(t − a2)(t − a3)
dt ∈ R,

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 aj and ak. We can see
that ai belong to γi and that these three curves
connect the corresponding ai with a common point
within ConvQ. Take a segment of γi connecting ai
with the common intersection point of all γ’s. Let
us denote the union of these three segments by ΓQ.
Then the support of the limiting root-counting mea-
sure μ coincides with ΓQ.

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 trans-
form [11]

Q(z)C′′ν (z) + Q
′(z)C′ν (z) +

Q′′(z)
8

Cν (z) +
Q′′′(z)

24
= 0.

In the case when Q(z) has all real zeros, the density
is explicitly given in [3].

The Cauchy transform Cν (z) and the logarithmic
potential potν (z) of a (complex-valued) measure ν
supported in C are given by:

Cν (z) =
∫
C

dν(ξ)
z − ξ

and

potν (z) =
∫
C

log |z − ξ| dν(ξ).

Cν (z) is analytic outside the support of ν [5].
In [11] we were able to find an additional proba-

bility measure ν which is easily described and from
which the measure μ is obtained by the inverse bal-
ayage, 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 lat-
ter one. This measure is uniquely determined by the
choice of a root of Q(z), and thus we in fact have con-
structed three different measures νi 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 se-
quence of monic (the leading coefficient is 1) Van
Vleck polynomials {Ṽn} converging to some monic
linear polynomial Ṽ . The existence of a linear poly-
nomial Ṽ is ensured by the existence of the limit of
the sequence of (unique) roots νn,in of {Ṽn}. The
above mentioned results guarantee the existence of
plenty of such converging sequences in ConvQ and
the limit ν̃ of these roots must necessarily belong to
ΓQ.

Having chosen {Ṽn} we take any sequence of the
corresponding {Sn,in}, deg Sn,in = n whose corre-
sponding sequence {Ṽn} has a limit. If we denote by
μn,in the root-counting measure of the correspond-
ing Stieltjes polynomial, we have proved that the se-
quence {μn,in } converges weakly to the unique prob-
ability measure μ

Ṽ
whose Cauchy transform C

Ṽ
(z)

satisfies the equation

C2
Ṽ

(z) =
Ṽ (z)
Q(z)

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 tra-
jectory joins two zeros of Q and the other one joins
the third zero of Q with the root of the limiting Van
Vleck polynomial.

3 Bispectral problems

Concerning the situation when k = 4 certain gen-
eral 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 situ-
ation 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 distribu-
tion of Stieltjes polynomials bears some resemblance
to the k = 3 case.

There are still several questions open. In addi-
tion, many other unsolved problems can be found for
higher linear differential equations with polynomial
coefficients.

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Acta Polytechnica Vol. 51 No. 4/2011

−5 0 5 10

−3 −2 −1 0 1 2
0

1

2

3

4

−3 −2 −1 0 1 2
0

1

2

3

4

Fig. 3: The left part: The location of roots for Q(x) = (x +5)(x +1)(x − 5)(x − 12), P = 0, and n = 6. The dots
have the same meaning as in Fig. 1. The right upper part: The union of roots of (quadratic) Van Vleck polynomials
for Q(z) = (z +1)(z − 2)(z − 2 − 4i)(z +3 − 2i), P = 0, and n = 20. The lower part: roots of a particular Stieltjes
polynomial (in red) and the roots of the corresponding Van Vleck polynomial (in green)

Acknowledgement

This work has been supported by the Czech Ministry
of Education, Youth and Sports within the project
LC06002 and by GACR grant P203/11/0701.

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Boris Shapiro
E-mail: shapiro@math.su.se
Department of Mathematics
Stockholm University,
SE-106 91 Stockholm, Sweden

Miloš Tater
E-mail: tater@ujf.cas.cz
Department of Theoretical Physics
Nuclear Physics Institute
AS CR v.v.i.
CZ-250 68 Řež, Czech Republic

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