Double confluent Heun function (HeunD)

Note

On heun.xyz, “HeunD” refers to canonical solutions of the double confluent Heun equation (DCHE) as developed in Ronveaux (ed.), Heun’s Differential Equations (1995), Part C.
Conventions vary across papers and CAS; the only robust way to translate is: match the differential equation (and the chosen normalization).

1. What “double confluent” means and why it matters

Heun’s general equation (four regular singular points) admits several confluent limits in which singularities merge. The double confluent limit is the most singular of the classical confluences: two pairs of regular singularities coalesce so that the resulting equation has

• exactly two singular points (at $z=0$ and $z=\infty$),
• and both are irregular (rank $1$ in the generic case).

This puts DCHE in the same conceptual family as equations studied with Stokes phenomena, connection coefficients, and Floquet theory. In physics, DCHE frequently appears after variable changes that convert a radial or angular ODE into something with exponential behavior at both ends, and it also contains classical periodic problems (e.g. Mathieu-type equations) as special cases.

A key structural feature: because the only finite singularity is at $z=0$, solutions are analytic on the punctured plane $\mathbb C^\*=\mathbb C\setminus\{0\}$, but generally multi-valued around $z=0$. It is therefore natural to treat solutions as living on the logarithm Riemann surface $\Omega$ (the universal cover of $\mathbb C^\*$).

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2. The symmetric canonical DCHE and its parameters

2.1 Canonical form used on this site

Following Ronveaux (Part C), we study DCHE in the symmetric canonical form

$$D^2y +\alpha\!\left(z+\frac1z\right)Dy +\left[ \left(\beta_1+\frac12\right)\alpha z +\left(\frac{\alpha^2}{2}-\gamma\right) +\left(\beta_{-1}-\frac12\right)\frac{\alpha}{z} \right]y =0,$$

where the Euler operator is

$$D \equiv z\frac{d}{dz}.$$

Parameters and their roles (heuristic but very useful):

• $\alpha$ is a singular (Stokes) parameter: it controls the leading exponential scales near $0$ and $\infty$ and determines Stokes rays.
• $\beta_{-1}$ and $\beta_1$ are also singular parameters: they control the leading algebraic powers in the canonical asymptotics near $z=0$ and $z=\infty$, respectively.
• $\gamma$ is the accessory parameter: it does not change the rank/type of the singularities, but it is the parameter that is typically quantized by global/monodromy/boundary conditions (so it often plays the role of an eigenvalue).

2.2 Ordinary-derivative form (if you want to compare to other conventions)

Using $Dy=z y'$ and $D^2y=z^2y''+zy'$, the same equation can be written as

$$z^2y''+\Bigl(z+\alpha(z^2+1)\Bigr)y' +\left[ \left(\beta_1+\frac12\right)\alpha z +\left(\frac{\alpha^2}{2}-\gamma\right) +\left(\beta_{-1}-\frac12\right)\frac{\alpha}{z} \right]y =0,$$

or divided by $z^2$,

$$y''+\left(\frac1z+\alpha\left(1+\frac1{z^2}\right)\right)y' +\left[ \left(\beta_1+\frac12\right)\frac{\alpha}{z} +\left(\frac{\alpha^2}{2}-\gamma\right)\frac1{z^2} +\left(\beta_{-1}-\frac12\right)\frac{\alpha}{z^3} \right]y =0.$$

This makes the “two irregular ends” explicit: the coefficients contain both positive and negative powers of $z$.

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3. Normal form and a fast route to WKB intuition

A standard trick (used systematically in Ronveaux) is to “gauge away” the first-derivative term.

For a general equation

$$D^2y+a(z)Dy+b(z)y=0,$$

one defines an invariant Laurent polynomial

$$B(z)\equiv b(z)-\left(\frac{a(z)}{2}\right)^2-\frac12\,D a(z).$$

After a suitable transformation $y=v(z)u(z)$ (with $v$ built from $a$), the equation becomes the normal form

$$D^2u+B(z)u=0.$$

For the symmetric DCHE, $a(z)=\alpha\left(z+\frac1z\right)$ and $b(z)$ is the bracketed coefficient above, and one finds the particularly transparent result

$$B(z)= -\frac{\alpha^2}{4}\left(z^2+\frac1{z^2}\right) +\alpha\beta_1 z +\alpha\beta_{-1}\frac1z -\gamma.$$

This normal form is extremely useful:

• The leading terms $-\frac{\alpha^2}{4}(z^2+z^{-2})$ explain immediately why both $z=0$ and $z=\infty$ are irregular of rank $1$ when $\alpha\neq0$.
• It is the right starting point for WKB/Stokes analysis and for connections to periodic ODEs.

3.1 Periodic-ODE viewpoint via $z=e^{ix}$

If you set $z=e^{ix}$, then $D=z\frac{d}{dz}=-i\frac{d}{dx}$, so $D^2=-\frac{d^2}{dx^2}$. The normal form becomes a second-order ODE with $2\pi$-periodic coefficients in $x$ (a Hill/Whittaker–Hill/Mathieu-type landscape depending on parameters). This is why Floquet theory is built into DCHE: the monodromy $z\mapsto e^{2\pi i}z$ is precisely $x\mapsto x+2\pi$.

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4. Singularities, Stokes sectors, and the two connection problems

Because $0$ and $\infty$ are irregular, the analytic theory splits into two intertwined “connection” questions:

1. Lateral connection (Stokes phenomenon): how asymptotic solutions in different angular sectors around the same irregular singularity are related (jumping across Stokes rays).
2. Central connection: how canonical bases near $z=\infty$ and near $z=0$ are related.

The book’s strategy is to define a canonical “seed” solution at $z\to\infty$, then generate other canonical solutions and all connection data using symmetry transformations and Wronskians.

A basic tool is the $D$-Wronskian

$$W[y_1,y_2](z)\equiv y_1Dy_2-y_2Dy_1.$$

For the symmetric DCHE it satisfies

$$W[y_1,y_2](z)=C\,e^{-\alpha z+\alpha/z}\qquad (z\in\mathbb C^\*),$$

with a constant $C$ depending only on the chosen pair of solutions.

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5. Canonical asymptotic solutions: a concrete definition of “HeunD”

The most operational way to define a special function in an irregular problem is: specify an asymptotic behavior in a Stokes sector, and prove uniqueness.

5.1 The canonical solution at infinity: $\Psi$

Ronveaux defines a distinguished solution $\Psi$ characterized by a simple algebraic asymptotic as $z\to\infty$ (within a suitable sector):

$$\Psi(\alpha,\beta_{-1},\beta_1,\gamma;z) =(\alpha z)^{-\beta_1-\frac12}\left(1+O\!\left(\frac1z\right)\right).$$

Moreover, $\Psi$ admits an asymptotic expansion

$$\Psi(\alpha,\beta_{-1},\beta_1,\gamma;z) \sim (\alpha z)^{-\beta_1-\frac12} \sum_{n=0}^\infty \psi_n(\alpha,\beta_{-1},\beta_1,\gamma)\,(\alpha z)^{-n}.$$

The coefficients $\psi_n$ are determined uniquely by the three-term recursion

$$(n+1)\psi_{n+1} -\Bigl((n+\beta_1+\tfrac12)^2-\gamma+\tfrac{\alpha^2}{2}\Bigr)\psi_n +\alpha^2(n+\beta_1-\beta_{-1})\psi_{n-1} =0, \qquad n\in\mathbb N,$$

with

$$\psi_0=1,\qquad \psi_{-1}=0.$$

Tip

If you only need accurate values for large $|z|$ in a fixed Stokes sector, this asymptotic expansion plus the recurrence is often the simplest “numerical definition” of the canonical HeunD solution.

5.2 A second canonical solution at infinity

A second independent solution at $z\to\infty$ has the characteristic exponential behavior

$$y_{\infty}^{(2)}(z)\sim e^{-\alpha z}(\alpha z)^{\beta_1-\frac12}\left(1+O\!\left(\frac1z\right)\right),$$

again sectorially (the exponential dominance depends on $\arg(\alpha z)$).

Together, the pair “algebraic” vs “exponentially scaled” is the natural rank-1 irregular analogue of a Frobenius basis at a regular singular point.

5.3 Canonical solutions at $z\to0$

There is a corresponding canonical pair as $z\to0$:

• one solution exhibits the essential exponential scale $e^{\alpha/z}$,
• the other is purely algebraic (up to sectorial corrections).

Their leading behaviors can be written as

$$y_{0}^{(1)}(z)\sim e^{\alpha/z}\left(\frac{\alpha}{z}\right)^{-\beta_{-1}-\frac12}\left(1+O(z)\right),$$

and

$$y_{0}^{(2)}(z)\sim \left(\frac{\alpha}{z}\right)^{\beta_{-1}-\frac12}\left(1+O(z)\right),$$

again understood sectorially because of Stokes phenomena.

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8. Floquet solutions and convergent Laurent expansions on $\mathbb C^\*$

A defining feature of DCHE (and one of the most useful computational handles) is the existence of solutions with controlled monodromy around $0$.

8.1 Monodromy operator and characteristic exponent

Let $m$ denote analytic continuation around $z=0$ once:

$$(my)(z)=y(e^{2\pi i}z).$$

A Floquet solution is a solution satisfying

$$y(e^{2\pi i}z)=e^{2\pi i\nu}\,y(z),$$

for some $\nu\in\mathbb C$ called the characteristic exponent (or Floquet exponent).

Because we are working on $\Omega$ (the log surface), the expression $z^\nu=e^{\nu\log z}$ is well-defined there.

8.2 Laurent-series representation

Ronveaux shows that a Floquet solution can be represented as

$$y(z)=z^\nu h(z),$$

where $h$ is holomorphic on $\mathbb C^\*$ and therefore has a convergent Laurent series

$$h(z)=\sum_{n=-\infty}^{\infty}\eta_n z^n \qquad (z\in\mathbb C^\*).$$

Substituting $y=z^\nu\sum_{n\in\mathbb Z}\eta_n z^n$ into the symmetric DCHE yields a three-term recurrence valid for all $n\in\mathbb Z$:

$$\alpha\left(n+\nu+\beta_{-1}+\frac12\right)\eta_{n+1} +\left((n+\nu)^2-\gamma+\frac{\alpha^2}{2}\right)\eta_n +\alpha\left(n+\nu+\beta_1-\frac12\right)\eta_{n-1} =0, \qquad n\in\mathbb Z.$$

This recurrence is central for:

• computing Floquet solutions numerically via continued fractions / minimal-solution conditions,
• studying when $\nu$ is integral or half-integral (single-valuedness or sign changes),
• and formulating eigenvalue problems in $\gamma$.

Caution

The recurrence is two-sided ($n\in\mathbb Z$). Picking a solution that converges/behaves correctly in both $n\to+\infty$ and $n\to-\infty$ is a nontrivial constraint and is exactly where Floquet/continued-fraction methods enter.

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9. Quasi-polynomials: when the “asymptotics” actually terminate

A striking phenomenon in DCHE is the existence of special solutions whose asymptotic series terminate in a way that produces a polynomial factor—analogous in spirit to Heun polynomials in the regular case.

Ronveaux calls these quasi-polynomials. A quasi-polynomial solution of degree $N$ has the form

$$y(z)=z^\nu \exp\!\left(-\alpha\left(\varepsilon_+ z+\varepsilon_-\frac1z\right)\right) P_N(z), \qquad z\in\Omega,$$

where

• $\nu\in\mathbb C$,
• $\varepsilon_\pm\in\{0,1\}$,
• and $P_N$ is a polynomial of degree $N$ with $P_N(0)\neq0$.

A particularly important subcase is the “pure polynomial” Floquet factor:

$$y(z)=z^\nu P_N(z).$$

Ronveaux shows that such a solution exists only under discrete constraints linking $(\beta_{-1},\beta_1)$ and a termination condition on the asymptotic coefficients $\psi_n$ of the canonical $\Psi$:

• one must have $\beta_{-1}-\beta_1=N+1$ (with $N\in\mathbb N^\*$),
• and the coefficient $\psi_{N+1}$ must vanish.

Since $\psi_{N+1}$ is (for fixed $\alpha,\beta_{-1},\beta_1$) a polynomial in $\gamma$ of degree $N+1$, the condition $\psi_{N+1}=0$ yields a finite set of candidate accessory parameters $\gamma$ that produce quasi-polynomials.

Tip

Quasi-polynomials are the DCHE analogue of “exactly solvable” parameter points: they convert a global analytic-connection problem into an algebraic condition in $\gamma$.

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10. DCHE as a Floquet eigenvalue problem

A powerful point of view—especially for physics—is to treat $\gamma$ (or a linear shift of it) as an eigenvalue, with the Floquet exponent $\nu$ playing the role of the quasi-momentum/Bloch phase.

Ronveaux formulates an eigenvalue problem using the operator

$$G(\beta)y := z\left(D+\beta_1+\frac12\right)y+\frac1z\left(D+\beta_{-1}-\frac12\right)y,$$

and considers

$$\left(D^2+\alpha\,G(\beta)-\lambda\right)y=0,$$

with the identification

$$\lambda=-\frac{\alpha^2}{2}+\gamma.$$

For fixed $(\nu,\alpha,\beta_{-1},\beta_1)$, the eigenvalues form a countable family $\lambda_\mu(\alpha,\beta)$ indexed by $\mu\in\nu+\mathbb Z$, and in the unperturbed case $\alpha=0$ one has

$$\lambda_\mu(0,\beta)=\mu^2.$$

Moreover, for small $|\alpha|$ (in an appropriate domain), $\lambda_\mu(\alpha,\beta)$ is holomorphic in $\alpha^2$ and admits a power series expansion

$$\lambda_\mu(\alpha,\beta)=\mu^2+\sum_{m=1}^\infty \lambda_{\mu m}(\beta)\,\alpha^{2m}.$$

The first perturbative coefficient is explicitly

$$\lambda_{\mu 1}(\beta)= -\frac12+\frac{2\beta_{-1}\beta_1}{4\mu^2-1}.$$

Note

This is one of the cleanest “graduate-student entry points” into DCHE: start from $\alpha=0$ where everything is elementary, then view DCHE as a deformation controlled by $\alpha$, with eigenvalues expanding in $\alpha^2$.

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11. A practical learning path (what to do first, second, third)

Here is a sequence that tends to work well if you’re learning DCHE for research use.

Step 1 — Master the canonical form and normal form

• Be completely comfortable moving between:
  • the symmetric canonical DCHE,
  • the ordinary-derivative form,
  • and the normal form $D^2u+B(z)u=0$ with explicit $B(z)$.

Step 2 — Work out asymptotics by hand once

• Derive the leading exponential scales near $z\to\infty$ and $z\to0$ (you will rediscover the $1$ vs $e^{-\alpha z}$ and $1$ vs $e^{\alpha/z}$ dichotomies).
• Compute the first few $\psi_n$ from the recurrence, and see explicitly how $\gamma$ enters.

Step 3 — Learn what “Floquet solutions” really mean in the $z$-plane

• Prove to yourself that analytic functions on $\mathbb C^\*$ have Laurent expansions.
• Use the recurrence for $\eta_n$ to understand why selecting a physically relevant solution is a two-sided minimal-solution problem.

Step 4 — Connect to periodic ODE intuition via $z=e^{ix}$

• Translate monodromy around $z=0$ into a Floquet condition under $x\mapsto x+2\pi$.
• Interpret $\nu$ as a quasi-momentum parameter when the problem becomes a Hill-type equation.

Step 5 — Study special solutions and spectral conditions

• Quasi-polynomials: understand termination conditions and how they quantize $\gamma$.
• Eigenvalue problems: understand why $\lambda=-\alpha^2/2+\gamma$ is a natural spectral parameter.

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12. Exercises (recommended)

1. Normal form derivation. Starting from the symmetric canonical DCHE, compute $B(z)=b-(a/2)^2-\frac12Da$ and verify

   $$B(z)= -\frac{\alpha^2}{4}\left(z^2+\frac1{z^2}\right) +\alpha\beta_1 z +\alpha\beta_{-1}\frac1z -\gamma.$$

2. Compute coefficients. Use the $\psi_n$ recurrence to compute $\psi_1,\psi_2,\psi_3$ explicitly and check how the degree in $\gamma$ grows.

3. Wronskian check. For the equation $D^2y+a(z)Dy+b(z)y=0$, derive the general identity

   $$W[y_1,y_2](z)=C\exp\!\left(-\int \frac{a(z)}{z}\,dz\right),$$

   and specialize to $a(z)=\alpha(z+1/z)$ to recover $e^{-\alpha z+\alpha/z}$.

4. Floquet-to-Laurent logic. Prove: if $y$ is Floquet with exponent $\nu$, then $h(z)=z^{-\nu}y(z)$ is single-valued on $\mathbb C^\*$ and therefore has a Laurent expansion.

5. Quasi-polynomial condition. In the subcase $y=z^\nu P_N(z)$, relate $\beta_{-1}-\beta_1=N+1$ to the balance of asymptotics at $0$ and $\infty$.

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References

• A. Ronveaux (ed.), Heun’s Differential Equations, Oxford University Press (1995). Part C: Double Confluent Heun Equation.