The Heun Confluence Hierarchy

The Heun hierarchy is not a list of increasingly complicated function names. It is the classification obtained when the four regular singularities of the general Heun equation merge in controlled groups. A two-point cluster generically creates a rank-$1$ irregular point, a three-point cluster a rank-$2$ point, and a four-point cluster a rank-$3$ point.

The word controlled is essential. Singular locations must move while parameters, coordinates, and sometimes gauges scale so that a higher polar moment survives. Merely setting two locations equal usually produces a degenerate equation, not the intended confluent class. This page turns that principle into a five-row atlas and verifies one complete limit.

Five collision partitions, not one ladder

Ignore the labels of the four original regular singularities and record only which ones have joined the same collision cluster. The five integer partitions of $4$ are

$$\begin{aligned} &1+1+1+1,\qquad 2+1+1,\\ &2+2,\qquad 3+1,\qquad 4. \end{aligned}$$

They give precisely the five standard Heun types:

Partition	Generic pattern	Class / label
$1+1+1+1$	$\mathrm R+\mathrm R+\mathrm R+\mathrm R$	GHE / HeunG
$2+1+1$	$\mathrm I_1+\mathrm R+\mathrm R$	CHE / HeunC
$2+2$	$\mathrm I_1+\mathrm I_1$	DCHE / HeunD
$3+1$	$\mathrm I_2+\mathrm R$	BHE / HeunB
$4$	$\mathrm I_3$	THE / HeunT

Here $\mathrm R$ denotes a regular singularity and $\mathrm I_r$ a generic unramified irregular singularity of rank $r$. This is the standard generic Heun confluence scheme: special coefficient cancellations can make a nominal regular point ordinary or lower an irregular slope.

The Heun confluence diamond. Each arrow merges one more collision block, but it represents a scaled limit of coefficients—not a collision of positions alone. The diagram classifies singularity patterns; it does not specify a unique parameter map or solution normalization.

The two middle partitions are incomparable: $2+2$ and $3+1$ are alternative coarsenings of $2+1+1$. Thus the doubly confluent and biconfluent equations are sibling branches, not successive rungs of one chain.

One DLMF convention fixes all five equations

Names such as “HeunC” do not by themselves specify an ODE. On this page all canonical equations use the DLMF forms. The same letters are reused by that source, so the parameters in one displayed equation are local to that equation; they do not pass unchanged through an arrow of the diagram.

General Heun: four regular points

The starting equation is

$$\begin{aligned} y'' &+ \left( \frac{\gamma}{z} + \frac{\delta}{z-1} + \frac{\epsilon}{z-a} \right)y'\\ &+ \frac{\alpha\beta z-q}{z(z-1)(z-a)}y =0, \end{aligned}$$

with

$$\gamma+\delta+\epsilon=\alpha+\beta+1.$$

The candidate singularities $0$, $1$, $a$, and $\infty$ are all regular, and generically none is removable. After Möbius normalization the equation has six independent canonical parameters: $a,q,\alpha,\beta,\gamma,\delta$.

Confluent Heun: two regular points and rank one

The DLMF confluent Heun equation is

$$\begin{aligned} y'' &+ \left( \frac{\gamma}{z} + \frac{\delta}{z-1} + \epsilon \right)y'\\ &+ \frac{\alpha z-q}{z(z-1)}y =0. \end{aligned}$$

The points $0$ and $1$ are generically regular singularities, while infinity is generically irregular of rank $1$. This canonical form has five parameters.

Doubly confluent Heun: rank one at both ends

The DLMF doubly confluent equation is

$$y'' + \left( \frac{\delta}{z^2} + \frac{\gamma}{z} + 1 \right)y' + \frac{\alpha z-q}{z^2}y =0.$$

The only singularities are $0$ and $\infty$; generically both are irregular of rank $1$. Generically there is no regular singular point at which to define a Frobenius-normalized local function. The canonical equation has four parameters.

Biconfluent Heun: one regular point and rank two

The DLMF biconfluent equation is

$$y'' - \left( \frac{\gamma}{z} + \delta + z \right)y' + \frac{\alpha z-q}{z}y =0.$$

The minus sign is part of this convention. The point $0$ is generically regular and infinity is generically irregular of rank $2$. This equation also has four parameters, but its singularity partition differs from the doubly confluent case.

Triconfluent Heun: one rank-three point

The DLMF triconfluent equation is

$$y'' + (\gamma+z)z\,y' + (\alpha z-q)y =0.$$

Every finite point is ordinary. The only singularity lies at infinity and is irregular of rank $3$. Three parameters remain after the canonical coordinate and gauge normalizations.

Class	Canonical parameter count	Finite regular points	Irregular points
GHE	$6$	$3$	none
CHE	$5$	$2$	$\infty$ of rank $1$
DCHE	$4$	none	$0,\infty$ of rank $1$
BHE	$4$	$1$	$\infty$ of rank $2$
THE	$3$	none; all finite points are ordinary	$\infty$ of rank $3$

The counts describe these canonical forms, not the dimension of every physical parameter space that can reduce to them.

Normal form reads off the generic rank

For

$$y''+p(z)y'+r(z)y=0,$$

the Liouville substitution

$$y = \exp\left( -\frac12\int^z p(s)\,\dd s \right)\psi$$

gives

$$\psi''+T(z)\psi=0, \qquad T=r-\frac12p'-\frac14p^2.$$

Near a finite point $t=0$, a generic unramified centered/projective rank-$r$ irregular type has

$$T(t) \sim \frac{c}{t^{2r+2}}, \qquad c\ne0.$$

More generally, if $T(t)\sim c\,t^{-m}$ with $m>2$, its centered, or projective, formal slope is $m/2-1$. Even $m$ gives the integer, unramified slopes displayed here; odd $m$ signals a ramified half-integer slope. For the five DLMF canonical equations on this page, this centered audit agrees with their stated generic ranks.

At infinity, inversion $w=1/z$ is Möbius and has zero Schwarzian. After the accompanying half-density rescaling that restores Liouville normal form, the transformed coefficient is

$$\widetilde T(w) = w^{-4}T(1/w).$$

Equivalently, centered/projective rank $r$ at infinity corresponds generically to

$$T(z)\sim c z^{2r-2}.$$

Applying this test gives a useful audit:

Type, point, rank	Leading $T$	Centered pair
GHE, $\infty$, $r=0$	$O(z^{-2})$	powers of $z$
CHE, $\infty$, $r=1$	$-\epsilon^2/4$	$\exp(\pm\epsilon z/2)$
DCHE, $0$, $r=1$	$-\delta^2/(4z^4)$	$\exp(\pm\delta/(2z))$
DCHE, $\infty$, $r=1$	$-1/4$	$\exp(\pm z/2)$
BHE, $\infty$, $r=2$	$-z^2/4$	$\exp(\pm z^2/4)$
THE, $\infty$, $r=3$	$-z^4/4$	$\exp(\pm z^3/6)$

The signs in the exponential pairs can be exchanged. The table reads the exponential difference. A common nonmeromorphic exponential twist can change the full $GL(2)$ Katz rank while leaving this centered data unchanged; a meromorphic scalar gauge cannot. Thus an arbitrary uncentered presentation still requires the full formal audit of Chapter 1.

The displayed ranks are generic

If a leading parameter vanishes, recompute the formal type. For example, $\delta=0$ removes the $z^{-4}$ term at the finite DCHE singularity. Similarly, the CHE becomes regular at infinity when both $\epsilon=0$ and $\alpha=0$. Intermediate degenerations can be ramified, so one should use formal slopes or Katz rank rather than infer an invariant rank from one raw companion matrix. Regular entries can also disappear: for example, $z=0$ is ordinary in the CHE or BHE when $\gamma=q=0$.

A complete general-to-confluent limit

The first arrow can be verified without guesswork. Fix target CHE parameters

$$(q,\alpha,\gamma,\delta,\epsilon), \qquad \epsilon\ne0,$$

and let the third finite Heun singularity be $a=A\to\infty$. In the general Heun equation choose

$$\begin{aligned} \epsilon_{\mathrm G} &=-A\epsilon,\\ \alpha_{\mathrm G} &=\frac{\alpha}{\epsilon},\\ \beta_{\mathrm G} &= -A\epsilon + \gamma+\delta-1 - \frac{\alpha}{\epsilon},\\ q_{\mathrm G} &=-Aq. \end{aligned}$$

The Fuchs relation holds exactly:

$$\gamma+\delta+\epsilon_{\mathrm G} = \alpha_{\mathrm G}+\beta_{\mathrm G}+1.$$

For fixed $z$ away from $0$ and $1$,

$$\frac{\epsilon_{\mathrm G}}{z-A} = \frac{\epsilon}{1-z/A} \longrightarrow \epsilon.$$

Moreover,

$$\alpha_{\mathrm G}\beta_{\mathrm G} = -A\alpha + \frac{\alpha}{\epsilon} \left( \gamma+\delta-1-\frac{\alpha}{\epsilon} \right),$$

so

$$\frac{ \alpha_{\mathrm G}\beta_{\mathrm G}z-q_{\mathrm G} }{ z(z-1)(z-A) } \longrightarrow \frac{\alpha z-q}{z(z-1)}.$$

Thus the general Heun coefficients converge locally uniformly on compact subsets of

$$\mathbb C\setminus\{0,1\}$$

to the CHE coefficients. With fixed, $A$-independent initial data at an ordinary base point, the solutions converge uniformly on compact subsets of any fixed simply connected domain avoiding $0$ and $1$ once $A$ lies outside it. Equivalently, one may state convergence along fixed continuation paths. This is standard continuous dependence on ODE coefficients.

What made the limit confluent was not only $A\to\infty$. The parameters $\epsilon_{\mathrm G}$, $\beta_{\mathrm G}$, and $q_{\mathrm G}$ scale linearly with $A$ so that an irregular polar moment survives; $q_{\mathrm G}$ diverges when $q\ne0$ and remains zero when $q=0$. If every general-Heun parameter were held fixed, the disappearing pole would contribute neither the constant $\epsilon$ in $p$ nor the linear numerator $\alpha z-q$ in the target equation.

The two middle branches encode different collisions

The diamond can now be read geometrically.

Confluent to doubly confluent

The CHE has one double cluster and two singleton clusters:

$$2+1+1.$$

Merging the two singleton clusters with a compensating parameter scaling creates a second rank-$1$ irregular point:

$$2+1+1 \longrightarrow 2+2.$$

The result has two irregular ends. Problems in this class naturally require sectorial data at both ends rather than a regular-to-irregular connection.

Confluent to biconfluent

Instead, merge one singleton into the existing double cluster:

$$2+1+1 \longrightarrow 3+1.$$

The original irregular point rises from rank $1$ to rank $2$, while one regular singularity remains. This is the biconfluent pattern, not the doubly confluent one.

Finally, merging the two blocks of either $2+2$ or $3+1$ gives the single four-point cluster $4$, hence the triconfluent rank-$3$ type. These are statements about collision partitions. Each concrete arrow still needs its own coordinate, parameter, and gauge scaling.

Equation class and named function are different data

The abbreviations GHE, CHE, DCHE, BHE, and THE classify equations. The software-style labels HeunG, HeunC, HeunD, HeunB, and HeunT select particular solutions only after a parameter convention and normalization have been declared.

Class	Geometry near $z=0$ in the DLMF chart	Natural local datum
GHE	Regular singular	A Frobenius branch, often analytic and normalized to $1$
CHE	Regular singular	A Frobenius branch, often analytic and normalized to $1$
DCHE	Irregular singular	Sectorial asymptotics; a CAS may instead normalize at an ordinary point
BHE	Regular singular	A Frobenius branch after fixing the equation convention
THE	Ordinary point	Two initial values determine an entire solution

This distinction is most acute for HeunD: in the two-irregular-point canonical picture, $z=0$ is not a place where “analytic and equal to $1$” defines a Frobenius solution. Different packages use different canonical charts and ordinary-point normalizations. The later parameter-crosswalk page will translate the house, DLMF, Maple, and Wolfram forms; until then, always write the ODE beside the function symbol.

What confluence preserves—and what it replaces

A controlled outer limit preserves the differential order and, after a fixed initial-value normalization, the two-dimensional solution space on common domains away from the collision. It does not preserve every familiar piece of Fuchsian data in its original form.

Before collision	After an irregular confluence
Frobenius powers at separate regular points	Formal exponentials, formal powers, and sectorial sums
Individual small loops	A loop around the cluster and sector-dependent Stokes factors
Local exponent differences	Irregular type plus formal monodromy data
Raw connection matrices between Frobenius bases	Renormalized limits that can produce Stokes matrices
A finite accessory coordinate $q_{\mathrm G}$	A scaled or shifted target accessory coordinate

The product of monodromies around a shrinking cluster can approach the full analytic monodromy of the irregular point, which factors into formal monodromy and Stokes matrices. A single raw local monodromy matrix does not generically become a single Stokes matrix. Chapter 1’s confluence analysis gives the required outer, inner, and basis-normalization ledger; the next page constructs the canonical local and asymptotic bases for the five Heun classes.

Common pitfalls

Drawing a linear hierarchy. The doubly confluent and biconfluent equations are different branches from the CHE. Their parameter counts happen to agree, but their singularity patterns do not.

Colliding positions without scaling coefficients. Bounded residues usually yield a regular or degenerate limit. An irregular type requires a higher polar moment to survive.

Treating the generic rank as universal. Vanishing leading coefficients can lower or ramify the formal type. Recompute $T$ or the invariant formal slopes on special parameter strata.

Carrying the same parameter names through an arrow. DLMF reuses $q,\alpha,\gamma,\delta,\epsilon$ across inequivalent equations. A confluence map generally shifts, rescales, or sends several of them to infinity.

Equating coefficient convergence with function convergence. Fixed initial-value solutions behave continuously on a common outer domain, but Frobenius or sectorial bases can diverge and require right renormalizations.

Calling a CAS symbol a canonical equation. A function name also hides a normalization, branch, and sometimes a different chart. State the ODE and initial or asymptotic data.

Exercises

1. Recover the five classes from partitions

List the integer partitions of $4$. Within the standard generic Heun confluence scheme, replace every block of size $m$ by an irregular point of rank $m-1$, with a size-$1$ block interpreted as a regular point. Match the five results to the Heun classes.

Solution

The partitions and patterns are

$$\begin{array}{c|c|c} 1+1+1+1 & \mathrm R+\mathrm R+\mathrm R+\mathrm R & \mathrm{GHE} \\ 2+1+1 & \mathrm I_1+\mathrm R+\mathrm R & \mathrm{CHE} \\ 2+2 & \mathrm I_1+\mathrm I_1 & \mathrm{DCHE} \\ 3+1 & \mathrm I_2+\mathrm R & \mathrm{BHE} \\ 4 & \mathrm I_3 & \mathrm{THE}. \end{array}$$

These are all possibilities because the integer partitions exhaust the unlabeled ways to group four objects.

2. Audit rank one at the confluent ends

For the DLMF CHE, compute $T$ through order $z^{-1}$ at infinity. For the DCHE, compute its leading normal coefficient at both $0$ and $\infty$. Recover all three rank-$1$ entries in the audit table.

Solution

For the CHE,

$$p = \epsilon+\frac{\gamma+\delta}{z}+O(z^{-2}), \qquad r = \frac{\alpha}{z}+O(z^{-2}).$$

Therefore

$$T_{\mathrm C}(z) = -\frac{\epsilon^2}{4} + \frac{ \alpha-\epsilon(\gamma+\delta)/2 }{z} + O(z^{-2}).$$

For $\epsilon\ne0$, the nonzero constant leading term gives rank $1$ at infinity and centered formal exponentials $\exp(\pm\epsilon z/2)$.

For the DCHE near $0$,

$$p = \frac{\delta}{z^2} + \frac{\gamma}{z} + 1, \qquad r = -\frac{q}{z^2} + \frac{\alpha}{z}.$$

Hence

$$T_{\mathrm D}(z) = -\frac{\delta^2}{4z^4} + \frac{ \delta(1-\gamma/2) }{z^3} + O(z^{-2}),$$

which gives rank $1$ and factors $\exp(\pm\delta/(2z))$ when $\delta\ne0$. At infinity,

$$T_{\mathrm D}(z) = -\frac14 + \frac{\alpha-\gamma/2}{z} + O(z^{-2}),$$

so the second end also has rank $1$, with factors $\exp(\pm z/2)$.

3. Audit ranks two and three

Put the DLMF BHE and THE in normal form. Recover rank $2$ for the biconfluent equation and rank $3$ for the triconfluent equation.

Solution

For the BHE,

$$p=-\frac{\gamma}{z}-\delta-z, \qquad r=\alpha-\frac qz.$$

Because $p'=-1+O(z^{-2})$,

$$T_{\mathrm B}(z) = -\frac{z^2}{4} -\frac{\delta z}{2} +O(1).$$

The relation $2\rho-2=2$ gives rank $\rho=2$, with centered exponentials $\exp(\pm z^2/4)$.

For the THE,

$$p=z^2+\gamma z, \qquad r=\alpha z-q.$$

Again the dominant contribution is $-p^2/4$, so

$$T_{\mathrm T}(z) = -\frac{z^4}{4} -\frac{\gamma z^3}{2} +O(z^2).$$

Because $2\rho-2=4$, the rank is $\rho=3$. Equivalently,

$$w^{-4}T(1/w) \sim -\frac1{4w^8},$$

whose pole order is $2\rho+2=8$. Integrating the leading square root gives the centered exponentials $\exp(\pm z^3/6)$.

4. Verify the general-to-confluent scaling

Using the parameter scaling in the worked limit, verify the Fuchs relation and both coefficient limits with $O(A^{-1})$ remainders.

Solution

The definitions first give

$$\beta_{\mathrm G}+1 = -A\epsilon+\gamma+\delta-\frac{\alpha}{\epsilon}.$$

Since $\alpha_{\mathrm G}=\alpha/\epsilon$, the Fuchs relation follows:

$$\alpha_{\mathrm G}+\beta_{\mathrm G}+1 = \gamma+\delta-A\epsilon = \gamma+\delta+\epsilon_{\mathrm G}.$$

For the first-derivative coefficient,

$$\frac{\epsilon_{\mathrm G}}{z-A} = \epsilon \left( 1+\frac zA+O(A^{-2}) \right).$$

If

$$C = \frac{\alpha}{\epsilon} \left( \gamma+\delta-1-\frac{\alpha}{\epsilon} \right),$$

then $\alpha_{\mathrm G}\beta_{\mathrm G}=-A\alpha+C$ and

$$\begin{aligned} & \frac{ \alpha_{\mathrm G}\beta_{\mathrm G}z-q_{\mathrm G} }{ z(z-1)(z-A) }\\ &\qquad= \frac{ \alpha z-q-Cz/A }{ z(z-1)(1-z/A) }\\ &\qquad= \frac{\alpha z-q}{z(z-1)} +O(A^{-1}) \end{aligned}$$

on compact sets away from $0$ and $1$.

5. Show why a moving point is insufficient

Let $a=A\to\infty$ in the GHE while all coefficients $q,\alpha,\beta,\gamma,\delta,\epsilon$ remain bounded. Determine the coefficientwise limit for fixed $z$.

Solution

For fixed $z$,

$$\frac{\epsilon}{z-A}\longrightarrow0$$

and

$$\frac{\alpha\beta z-q}{z(z-1)(z-A)} = O(A^{-1}).$$

The limit is therefore

$$y'' + \left( \frac{\gamma}{z} + \frac{\delta}{z-1} \right)y' =0.$$

It is a degenerate Fuchsian equation, not the generic CHE. The constant term in the CHE first-derivative coefficient and the nonzero target numerator require the divergent scaling used on the page.

6. Locate the hypergeometric boundary of the CHE

Set $\epsilon=0$ and $\alpha=0$ in the DLMF CHE. Show that infinity becomes regular and identify Gauss parameters $A_{\mathrm h},B_{\mathrm h},C_{\mathrm h}$.

Solution

The equation becomes

$$y'' + \left( \frac{\gamma}{z} + \frac{\delta}{z-1} \right)y' - \frac{q}{z(z-1)}y =0.$$

Now $p=O(z^{-1})$ and $r=O(z^{-2})$, so infinity is regular. Comparing with the Gauss equation gives

$$C_{\mathrm h}=\gamma,$$ $$A_{\mathrm h}+B_{\mathrm h}+1 = \gamma+\delta,$$

and

$$A_{\mathrm h}B_{\mathrm h}=-q.$$

Only setting $\epsilon=0$ is not sufficient in general: if $\alpha\ne0$, then $r\sim\alpha/z$ and infinity remains irregular, though the leading formal type is degenerate from the generic unramified CHE row.

7. Choose the correct starting basis

For generic canonical parameters in each of GHE, CHE, DCHE, BHE, and THE, decide whether $z=0$ naturally supports a Frobenius basis, a sectorial asymptotic basis, or an ordinary initial-value basis in the DLMF chart.

Solution

• Generically, GHE, CHE, and BHE have a regular singularity at $0$, so Frobenius bases are natural, with the usual resonant caveats.
• For $\delta\ne0$, DCHE has an irregular singularity at $0$, so its canonical local objects are formal and sectorial asymptotic bases. A software function may use an ordinary point elsewhere for normalization.
• THE has no finite singularity. The point $0$ is ordinary, and two initial values determine an entire solution.

This classification concerns the equation. A symbol such as HeunD or HeunT still requires a package-specific parameter and normalization declaration. On loci such as $\delta=0$ for DCHE, or after removable-pole cancellations in the other families, the local type must be recomputed.

References

• NIST DLMF equation 31.2.1 gives the general Heun equation; §31.2(i) gives its exponent and six-parameter convention. The four confluent forms and their singularity ranks are 31.12.1, 31.12.2, 31.12.3, and 31.12.4. Equation 31.12.3 is the corrected BHE form.
• A. Ronveaux, ed., Heun’s Differential Equations, Oxford University Press (1995), book record and DOI: CHE pp. 89–97, DCHE pp. 131–143, BHE pp. 193–198, THE pp. 253–257, and the classification and confluence addenda pp. 305–317.
• S. Yu. Slavyanov and W. Lay, Special Functions: A Unified Theory Based on Singularities, Oxford University Press (2000), Chapter 3, pp. 97–162, organizes the Heun class by singularity structure.
• A. Decarreau, M.-Cl. Dumont-Lepage, P. Maroni, A. Robert, and A. Ronveaux, “Formes canoniques des équations confluentes de l’équation de Heun,” Ann. Soc. Sci. Bruxelles Sér. I 92 (1–2) (1978), 53–78; A. Decarreau, P. Maroni, and A. Robert, “Sur les équations confluentes de l’équation de Heun,” 92 (3) (1978), 151–189. DLMF §31.12 cites these papers for the classification.
• K. Heun, “Zur Theorie der Riemann’schen Functionen zweiter Ordnung mit vier Verzweigungspunkten,” Math. Ann. 33 (1889), 161–179, doi:10.1007/BF01443849.