HeunG (General Heun Function)

Note

This page is about the general (non-confluent) Heun equation and its standard local solution HeunG (a.k.a. $Hl$). Confluent variants such as HeunC, HeunB, etc., are not covered here.

What HeunG is

The general Heun equation is the most general second-order Fuchsian ordinary differential equation with four regular singular points. After a Möbius transformation, these singularities can be placed at

$$z = 0,\; 1,\; a,\; \infty,\qquad a\neq 0,1.$$

The function commonly called HeunG in computer algebra systems is the normalized Frobenius solution around $z=0$ corresponding to the exponent $0$:

• it is analytic at $z=0$,
• it satisfies $y(0)=1$,
• it depends on an accessory parameter $q$ that controls global analytic behavior and typically becomes an eigenvalue in boundary value problems.

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1. Canonical general Heun equation and parameters

1.1 Canonical form

The canonical (general) Heun equation is commonly written as

$$\frac{d^2y}{dz^2} +\left(\frac{\gamma}{z}+\frac{\delta}{z-1}+\frac{\epsilon}{z-a}\right)\frac{dy}{dz} +\frac{\alpha\beta\, z-q}{z(z-1)(z-a)}\,y=0,$$

with complex parameters $(a,q,\alpha,\beta,\gamma,\delta,\epsilon)$ and with

$$a\neq 0,1.$$

The parameters are not independent: Fuchsian balance at infinity implies

$$\gamma+\delta+\epsilon=\alpha+\beta+1, \qquad\text{equivalently}\qquad \epsilon=\alpha+\beta-\gamma-\delta+1.$$

1.2 Singularities and local exponents

The equation is Fuchsian with regular singularities at

$$z=0,\;1,\;a,\;\infty.$$

A convenient way to summarize local exponents is via the Riemann $P$-symbol:

$$P\left\{ \begin{matrix} 0&1&a&\infty\\ 0&0&0&\alpha\\ 1-\gamma&1-\delta&1-\epsilon&\beta \end{matrix} \;\middle|\; z \right\},$$

meaning that the pairs of characteristic exponents are

• at $z=0$: $0,\;1-\gamma$,
• at $z=1$: $0,\;1-\delta$,
• at $z=a$: $0,\;1-\epsilon$,
• at $z=\infty$: $\alpha,\;\beta$.

1.3 What do $a$ and $q$ mean?

• $a$ locates the third finite singular point (in addition to $0$ and $1$). It is often called the singularity parameter.
• $q$ is the accessory parameter. Unlike $(\alpha,\beta,\gamma,\delta,\epsilon)$ (which fix exponent data), $q$ is not determined by local exponents; in applications it frequently plays the role of an eigenvalue fixed by global boundary/regularity conditions.

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2. Definition of the general Heun function HeunG

2.1 Local (Frobenius) solution at the origin

Among the two local Frobenius solutions about $z=0$, the one corresponding to exponent $0$ is analytic at $z=0$ and can be normalized by setting $y(0)=1$. This normalized local solution is what most CAS call the general Heun function:

Definition (local):
HeunG(a,q,α,β,γ,δ;z) is the solution of the canonical Heun equation that is analytic at $z=0$ and satisfies

$$\mathrm{HeunG}(a,q,\alpha,\beta,\gamma,\delta;0)=1.$$

In the Ronveaux (1995) convention, this same object is often denoted

$$Hl(a,q;\alpha,\beta,\gamma,\delta;z),$$

with $\epsilon$ understood implicitly through $\epsilon=\alpha+\beta-\gamma-\delta+1$.

Note

You will see multiple notations in the literature:

• local solution analytic at one singular point: often $Hl$ (or “local Heun function”),
• Heun function relative to two singularities (analytic at both): often $Hf$ with extra labels.

Most software symbols named HeunG refer to the local solution at $z=0$ (analytic at $0$, normalized to 1).

2.2 The second local solution at $z=0$

Generically (when $\gamma\notin\mathbb{Z}$), the second local solution behaves like

$$y(z)\sim z^{\,1-\gamma}\quad(z\to 0).$$

When $\gamma\in\mathbb{Z}$, the exponent difference $1-\gamma$ is an integer and the second local solution may involve a $\log z$ term (“logarithmic case”). This is one of the main sources of special-function subtlety for Heun equations.

2.3 Quickstart in CAS (Maple and Wolfram Language)

Most researchers first use HeunG through a CAS. The two most common interfaces are:

Maple

HeunG(a, q, alpha, beta, gamma, delta, z);
HeunGPrime(a, q, alpha, beta, gamma, delta, z);  # derivative w.r.t. z

Wolfram Language / Mathematica

HeunG[a, q, α, β, γ, δ, z]
HeunGPrime[a, q, α, β, γ, δ, z]   (* derivative w.r.t. z *)

Both systems use the same parameter order $(a,q,\alpha,\beta,\gamma,\delta;z)$ and the same canonical differential equation in §1, with the same normalization $y(0)=1$.

A good habit (especially when inheriting conventions from a paper) is to ask the CAS for its own definition:

• Maple: FunctionAdvisor(definition, HeunG);
• Wolfram Language: consult the built-in HeunG documentation page.

Caution

Many papers use different parameter names and sometimes swap the roles of $(\alpha,\beta)$ or rename the accessory parameter $q$. Always match your ODE to the canonical form in §1 before trusting a HeunG(...) call.

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3. Power-series expansion: coefficients and recurrence

3.1 Series ansatz and normalization

For the exponent-$0$ local solution about $z=0$, use the series

$$y(z)=\sum_{r=0}^{\infty}c_r z^r,\qquad c_0\neq 0,$$

and impose the standard normalization

$$c_0=1.$$

3.2 Three-term recurrence (Ronveaux 1995)

Substituting the series into the Heun equation yields a three-term recurrence. The starting relation is

$$-q\,c_0 + a\gamma\,c_1 = 0 \qquad\Rightarrow\qquad c_1=\frac{q}{a\gamma}\quad(\gamma\neq 0).$$

For $r\ge 1$,

$$P_r\,c_{r-1} - (Q_r+q)\,c_r + R_r\,c_{r+1}=0,$$

where

$$P_r=(r-1+\alpha)(r-1+\beta),$$ $$Q_r=r\Big[(r-1+\gamma)(1+a)+a\delta+\epsilon\Big],$$ $$R_r=(r+1)(r+\gamma)a,$$

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

Equivalently, the forward form is

$$c_{r+1}=\frac{(Q_r+q)c_r - P_r c_{r-1}}{R_r}\qquad(r\ge 1),$$

with the convention $c_{-1}=0$.

Caution

The recurrence assumes that the exponent difference at $z=0$ is not an integer obstruction; in particular, one commonly assumes

$$\gamma\neq 0,-1,-2,\dots$$

to avoid degeneracy of the Frobenius construction at $z=0$.

3.3 Radius of convergence

The power series about $z=0$ converges up to the nearest singularity. Since the finite singularities other than $0$ are at $z=1$ and $z=a$,

$$\text{radius of convergence} = \min(1,|a|).$$

So:

• if $|a|>1$, the disk of convergence is typically $|z|<1$,
• if $|a|<1$, the disk is typically $|z|<|a|$.

This matters in practice: when $a$ is close to $0$ or $1$, the local series can have a very small domain of convergence.

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4. The second local solution at $z=0$

A standard way to write the second independent local solution at $z=0$ (generic case) is

$$y_2(z)=z^{1-\gamma}\,\tilde y(z),$$

where $\tilde y(z)$ is again a local Heun function with shifted parameters.

One explicit parameter shift is

$$\alpha\mapsto \alpha+1-\gamma,\quad \beta\mapsto \beta+1-\gamma,\quad \gamma\mapsto 2-\gamma,\quad \delta\mapsto \delta,\quad \epsilon\mapsto \epsilon,$$

and

$$q\mapsto q_{II}=q+(a\delta+\epsilon)(1-\gamma).$$

So a local basis near $z=0$ is

$$y_1(z)=Hl(a,q;\alpha,\beta,\gamma,\delta;z),$$ $$y_2(z)=z^{1-\gamma}\,Hl(a,q_{II};\alpha+1-\gamma,\beta+1-\gamma,2-\gamma,\delta;z).$$

This is the direct analog of the hypergeometric basis ${}_2F_1$ and $z^{1-c}{}_2F_1$.

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5. Transformations

Heun theory is transformation-heavy. Two families are particularly important:

1. Möbius transformations of $z$ (permuting $\{0,1,a,\infty\}$).
2. Prefactor transformations of $y$ of the form $y=z^{\rho}(z-1)^{\sigma}(z-a)^{\tau}\tilde y$ to swap exponent choices.

Together these generate many equivalent local series representations (often summarized as “$192$ series”), but they reduce to the expected eight local solutions: two around each singularity.

5.1 A simple scaling identity

One especially useful identity relates expansions at $a$ and $1/a$:

$$Hl(a,q;\alpha,\beta,\gamma,\delta;z) = Hl\!\left(\frac{1}{a},\frac{q}{a};\alpha,\beta,\gamma,\alpha+\beta-\gamma-\delta+1;\frac{z}{a}\right).$$

This can be used to move $a$ inside or outside the unit disk to improve convergence regions.

5.2 An example where the accessory parameter transforms nontrivially

For the change of variables $\zeta = z/(z-1)$, one finds an identity of the form

$$Hl(a,q;\alpha,\beta,\gamma,\delta;z) = Hl\!\left(\frac{a}{a-1},\frac{q+\alpha\gamma a}{a-1};\alpha,\alpha-\delta+1,\gamma,\alpha+\beta+1;\frac{z}{z-1}\right).$$

The point to internalize is not the specific map, but the phenomenon: $q$ transforms in a genuinely non-obvious way, and it encodes global information.

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6. Hypergeometric degenerations

The general Heun equation can degenerate to the Gauss hypergeometric equation in special parameter limits. If $F(a,q;\alpha,\beta,\gamma,\delta;z)$ denotes the normalized exponent-$0$ Frobenius solution at $z=0$ with $F(0)=1$, then three useful degenerations are:

1. If $a=1$ and $q=\alpha\beta$,

$$F(1,\alpha\beta;\alpha,\beta,\gamma,\delta;z) = {}_2F_1(\alpha,\beta;\gamma;z).$$

2. If $a=0$ and $q=0$,

$$F(0,0;\alpha,\beta,\gamma,\delta;z) = {}_2F_1\!\big(\alpha,\beta;\alpha+\beta-\delta+1;z\big).$$

3. If $\epsilon=0$ and $q=a\alpha\beta$ (equivalently $\delta=\alpha+\beta-\gamma+1$),

$$F(a,a\alpha\beta;\alpha,\beta,\gamma,\alpha+\beta-\gamma+1;z) = {}_2F_1(\alpha,\beta;\gamma;z).$$

These limits are excellent sanity checks for both symbolic manipulations and numerics.

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7. Global solutions and why $q$ becomes an eigenvalue

A local HeunG/$Hl$ solution is defined near one singularity (e.g. near $z=0$). In contrast, the reference uses the term Heun function (denoted $Hf$) for a solution that is simultaneously a Frobenius solution about two singularities $s_1,s_2$, i.e. analytic in a region containing both (with branch cuts chosen for single-valuedness).

In many physical problems you impose two endpoint conditions such as:

• regularity at $z=0$ and $z=1$,
• ingoing behavior at one singular point and normalizability at another,
• prescribed monodromy around a contour.

These global requirements typically hold only for a discrete set of $q$ values. That is why $q$ behaves as an eigenvalue.

A practical picture: connection matching

To enforce “regular at $z=0$ and $z=1$” you can:

1. Build a local basis near $z=0$: $y_1=Hl(\cdots)$ and $y_2=z^{1-\gamma}Hl(\cdots)$.

2. Build a local basis near $z=1$ (by a transformation $z\mapsto 1-z$ or a dedicated local expansion).

3. Evaluate both bases and their derivatives at a matching point $z_*$ in an overlap region.

4. Solve for the connection coefficients and impose that the coefficient of the singular basis element vanishes.

This yields a scalar condition on $q$, i.e. an eigenvalue equation.

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8. Heun polynomials and series termination

A Heun polynomial is a solution that is simultaneously Frobenius at three singularities and has the structured form

$$Hp(z)=z^{\sigma_1}(z-1)^{\sigma_2}(z-a)^{\sigma_3}\,p(z),$$

where $p(z)$ is a polynomial and each $\sigma_i$ is one of the two local exponents at the corresponding singularity.

8.1 The simplest termination mechanism

From the recurrence, if

$$\alpha=-n,\qquad n\in\{0,1,2,\ldots\},$$

then $P_{n+1}=0$. If one chooses $q$ so that $c_{n+1}=0$, the series terminates and produces a polynomial of degree $n$ (class I in the reference). The admissible $q$ values satisfy an algebraic equation of degree $n+1$.

8.2 Eight classes of Heun polynomials

The reference classifies Heun polynomials into eight classes I–VIII, determined by the choices of $(\sigma_1,\sigma_2,\sigma_3)$ and corresponding constraints on $(\alpha,\beta)$.

A convenient summary is:

$$\begin{array}{c|ccc|cc} \text{Class} & \sigma_1 & \sigma_2 & \sigma_3 & \alpha & \beta\\ \hline \text{I} & 0 & 0 & 0 & -n & \gamma+\delta+\epsilon+n-1\\ \text{II} & 1-\gamma & 0 & 0 & \gamma-n-1 & \delta+\epsilon+n\\ \text{III} & 0 & 1-\delta & 0 & \delta-n-1 & \epsilon+\gamma+n\\ \text{IV} & 1-\gamma & 1-\delta & 0 & \gamma+\delta-n-2 & \epsilon+n+1\\ \text{V} & 0 & 0 & 1-\epsilon & \epsilon-n-1 & \gamma+\delta+n\\ \text{VI} & 1-\gamma & 0 & 1-\epsilon & \gamma+\epsilon-n-2 & \delta+n+1\\ \text{VII} & 0 & 1-\delta & 1-\epsilon & \delta+\epsilon-n-2 & \gamma+n+1\\ \text{VIII} & 1-\gamma & 1-\delta & 1-\epsilon & \gamma+\delta+\epsilon-n-3 & n+2 \end{array}$$

Note

Only class I are “polynomials” in the strict sense (single-valued polynomials in $z$). The other classes involve branch factors such as $z^{1-\gamma}$ or $(z-1)^{1-\delta}$ and require branch cuts for single-valuedness.

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9. Branch cuts and multi-valuedness

Even though the local series defines an analytic function in its convergence disk, analytic continuation around singularities produces monodromy, so the global object is generally multi-valued.

9.1 What software does

Computer algebra systems choose a principal branch, which is implemented by placing branch cuts that connect singular points to infinity.

In Wolfram Language, the principal branch of HeunG[a,q,α,β,γ,δ,z] has two branch cut discontinuities in the complex $z$-plane: one running from $z=1$ to infinity and one running from $z=a$ to infinity in the direction of $a$.

9.2 Researcher workflow

• If you are doing numerics, always record:
  • which branch you intend (path of continuation),
  • where you expect discontinuities,
  • how you validate (e.g. by checking the ODE numerically).
• If you are doing spectral/eigenvalue problems, make sure the boundary/regularity conditions are imposed on a consistent branch.

Caution

Many “mysterious” numerical issues with Heun functions are actually branch issues. If you see discontinuities or inconsistent values, check whether your evaluation path crosses a branch cut.

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10. How to reduce a given ODE to HeunG form

Suppose you start with a second-order linear ODE

$$w''(x)+p(x)w'(x)+r(x)w(x)=0,$$

and you suspect it has four regular singular points.

A robust reduction procedure is:

Step 1: map the singular points to $\{0,1,a,\infty\}$

If the finite regular singular points are at $x=x_1,x_2,x_3$ (distinct), choose a Möbius transformation

$$z=\frac{(x-x_1)(x_2-x_3)}{(x-x_3)(x_2-x_1)}$$

so that

$$x=x_1\mapsto z=0,\qquad x=x_2\mapsto z=1,\qquad x=x_3\mapsto z=a,$$

for some computed $a\neq 0,1$.

Step 2: normalize exponents with a gauge factor

Write

$$w(x(z)) = z^{\sigma_0}(z-1)^{\sigma_1}(z-a)^{\sigma_a}\,y(z),$$

and choose $(\sigma_0,\sigma_1,\sigma_a)$ so that $y(z)$ has the desired local exponents at $z=0,1,a$ (often one chooses the exponent $0$ at each finite singularity for the canonical Heun normalization).

Step 3: match coefficients to read off $(\gamma,\delta,\epsilon)$ and $(\alpha\beta,q)$

After Steps 1–2, bring the equation to the canonical form

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

Then:

• the residues of the $y'$ coefficient at $z=0,1,a$ give $(\gamma,\delta,\epsilon)$,
• the numerator of the $y$ coefficient gives $\alpha\beta$ (coefficient of $z$) and $q$ (minus the constant term).

Finally, enforce the Fuchs relation $\epsilon=\alpha+\beta-\gamma-\delta+1$ to determine $\epsilon$ (or as a consistency check).

Note

In practice, it is often easiest to do Step 3 symbolically in a CAS: expand the rational coefficient functions into partial fractions and match term-by-term.

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11. Common pitfalls

• Forgetting the constraint $\epsilon=\alpha+\beta-\gamma-\delta+1$ (software often hides $\epsilon$).
• Using forbidden values $a=0$ or $a=1$ (singularities collide; you enter a confluent regime).
• Logarithmic cases ($\gamma\in\mathbb{Z}$, similarly at $z=1$ or $z=a$ when $\delta$ or $\epsilon$ are integers): the naive Frobenius basis changes.
• Small radius of convergence when $|a|\ll 1$ or $a\approx 1$: use transformations or re-expansion.
• Branch cut crossings in analytic continuation: results can jump between branches.
• Assuming symmetry in $(\alpha,\beta)$: the equation is symmetric under $\alpha\leftrightarrow\beta$, but your chosen solution/branch and normalization may not make that symmetry manifest.
• Underestimating precision needs near singular points: high-precision arithmetic is often necessary.

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References

• Primary convention source for this page:
  A. Ronveaux (ed.), Heun’s Differential Equations, Oxford University Press (1995), Part A: “Heun’s equation” (canonical form, local solution $Hl$, transformations, Heun functions $Hf$, Heun polynomials $Hp$).

• Additional standard references you may see in the literature:

  • A. F. Slavyanov and W. Lay, Special Functions: A Unified Theory Based on Singularities (useful for broader special-function context).
  • R. S. Maier, work on the 192 solutions and transformation theory (useful for symmetry and connection problems).

Tip

If you use software implementations of HeunG, always check the exact differential equation and parameter convention (especially the definition of $q$) against the canonical form stated on this page.