HeunC (Confluent Heun Function)

Note

This page focuses on the confluent Heun equation (CHE) and the standard local confluent Heun function commonly called HeunC in computer algebra systems.

What HeunC is

The general Heun equation has four regular singular points. If two of those singularities coalesce (a confluence limit), you obtain the confluent Heun equation:

• two regular singular points (typically at $z=0$ and $z=1$),
• one irregular singular point at $z=\infty$ (rank 1 in the canonical CHE),
• an accessory parameter that is not determined by local exponent data and often becomes an eigenvalue under global boundary conditions.

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Conventions used on this page

Always state the equation you mean

There are multiple incompatible parameterizations called “confluent Heun equation” in the literature and in CAS.
To be unambiguous, always write the ODE and the normalization (e.g. $y(0)=1$) alongside the symbol HeunC.

To minimize confusion, this page uses three parameter sets and always labels them:

• DLMF / Wolfram parameters: $(q,\alpha,\gamma,\delta,\epsilon)$.
• Ronveaux (1995) parameters (Part B): $(p,\alpha_R,\gamma,\delta,\sigma)$.
• Maple parameters: $(\alpha_M,\beta_M,\gamma_M,\delta_M,\eta_M)$.

The same letters appear in different roles across the literature; the subscripts $R$ and $M$ are meant to keep you sane.

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1. Canonical confluent Heun equation

1.1 DLMF / Wolfram canonical form

A widely used canonical CHE (used by DLMF and the Wolfram Language) is

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

with complex parameters $(q,\alpha,\gamma,\delta,\epsilon)$ and independent variable $z$.

Multiplying by $z(z-1)$ gives the equivalent polynomial-coefficient form:

$$z(z-1)y'' + \big(\gamma(z-1)+\delta z+\epsilon\,z(z-1)\big)y' + (\alpha z-q)\,y = 0.$$

Singularities. For $\epsilon\neq 0$, this equation has

• regular singularities at $z=0$ and $z=1$,
• an irregular singularity of rank $1$ at $z=\infty$.

If $\epsilon=0$, then $z=\infty$ becomes a regular singularity, and the equation reduces (after standard identification) to the Gauss hypergeometric class (a Fuchsian ODE with three regular singularities).

1.2 Local exponents at $z=0$ and $z=1$

A quick Frobenius analysis gives characteristic exponents:

• at $z=0$: $0$ and $1-\gamma$,
• at $z=1$: $0$ and $1-\delta$.

So, when $\gamma\notin\mathbb{Z}$, two independent local solutions about $z=0$ behave like

$$y_1(z)\sim 1,\qquad y_2(z)\sim z^{1-\gamma}\quad(z\to 0).$$

Analogously at $z=1$ with $\delta$.

Resonant/logarithmic cases

When $\gamma\in\mathbb{Z}$ (or $\delta\in\mathbb{Z}$ at $z=1$), the exponent difference is an integer and the second local solution may involve $\log z$ (or $\log(1-z)$).
This is where different CAS conventions can diverge, and the normalization of HeunC needs extra care.

1.3 What do the parameters mean?

• $\gamma$ and $\delta$ control the exponent differences at $z=0$ and $z=1$.
• $\epsilon$ controls the strength of the irregular singularity at $\infty$ (it is the parameter that disappears in the hypergeometric limit).
• $q$ is the accessory parameter: it is not fixed by local exponent data and in applications is frequently determined by a global boundary condition (often an eigenvalue problem).
• $\alpha$ is the remaining parameter in the coefficient of $y$; together with $(q,\epsilon)$ it controls the asymptotics at infinity.

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2. Definition of the confluent Heun function HeunC

2.1 Wolfram/DLMF normalization

In the Wolfram Language, HeunC[q,α,γ,δ,ϵ,z] denotes the solution of the canonical CHE that is regular at $z=0$ and satisfies

$$\mathrm{HeunC}(q,\alpha,\gamma,\delta,\epsilon;0)=1.$$

This is precisely the local Frobenius solution with exponent $0$ at $z=0$.

Wolfram also notes:

• HeunC has a branch cut in the complex $z$-plane running from $z=1$ to $z=\infty$ (principal branch choice),
• values in “logarithmic cases” (notably nonpositive integer $\gamma$) are not determined in the same way as generic parameters.

2.2 Ronveaux (1995) canonical form and notation

Part B of Heun’s Differential Equations (Ronveaux, ed., 1995) works with the non-symmetrical canonical form (their Eq. (1.2.27)):

$$w''(z) +\left(4p+\frac{\gamma}{z}+\frac{\delta}{z-1}\right)w'(z) +\frac{4p\,\alpha_R\,z-\sigma}{z(z-1)}\,w(z)=0,$$

with parameters $(p,\alpha_R,\gamma,\delta,\sigma)$.

They introduce the Frobenius solution at the origin,

$$Hc^{(a)}(p,\alpha_R,\gamma,\delta,\sigma;z),\qquad Hc^{(a)}(\cdots;0)=1,$$

and (for suitable $p$) a solution defined by behavior at infinity (a “radial” solution).

About the superscripts

Ronveaux uses the superscripts $(a)$ and $(r)$ for “angular” and “radial” families (terminology inherited from spheroidal problems).

2.3 Converting between Ronveaux and Wolfram/DLMF

Comparing the two canonical equations gives the direct identification:

$$\epsilon = 4p,\qquad \alpha = 4p\,\alpha_R = \epsilon\,\alpha_R,\qquad q = \sigma,$$

with $\gamma$ and $\delta$ the same in both forms.

So, up to relabeling,

$$Hc^{(a)}(p,\alpha_R,\gamma,\delta,\sigma;z) \equiv \mathrm{HeunC}(q,\alpha,\gamma,\delta,\epsilon;z) \quad\text{with}\quad (q,\alpha,\epsilon)=(\sigma,4p\alpha_R,4p).$$

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3. Power-series definition and the three-term recurrence

The most important practical fact about HeunC is that it is defined by a convergent power series around $z=0$ with a three-term recurrence for its coefficients.

3.1 Series about $z=0$ (Ronveaux form)

Ronveaux defines the Frobenius solution at $z=0$ by

$$Hc^{(a)}(p,\alpha_R,\gamma,\delta,\sigma;z)=\sum_{k=0}^{\infty} c_k^{(a)}\, z^k, \qquad c_0^{(a)}=1.$$

The coefficients satisfy the three-term recurrence (their Eq. (2.2.14))

$$f_k^{(a)}\,c_{k+1}^{(a)} + g_k^{(a)}\,c_k^{(a)} + h_k^{(a)}\,c_{k-1}^{(a)} = 0, \qquad c_{-1}^{(a)}=0,$$

with

$$f_k^{(a)} = -(k+1)(k+\gamma),$$ $$g_k^{(a)} = k\big(k-4p+\gamma+\delta-1\big)-\sigma,$$ $$h_k^{(a)} = 4p\,(k+\alpha_R-1).$$

3.2 The same recurrence in $(q,\alpha,\gamma,\delta,\epsilon)$ form

Using $\epsilon=4p$, $\alpha=\epsilon\alpha_R$, $q=\sigma$, the recurrence can be written as

$$-(k+1)(k+\gamma)\,c_{k+1} +\big(k(k-\epsilon+\gamma+\delta-1)-q\big)c_k +\big(\epsilon(k-1)+\alpha\big)c_{k-1}=0,$$

with $c_{-1}=0$, $c_0=1$.

First coefficient. Setting $k=0$ gives

$$c_1 = -\frac{q}{\gamma},$$

so

$$\mathrm{HeunC}(q,\alpha,\gamma,\delta,\epsilon;z)=1-\frac{q}{\gamma}z+O(z^2),$$

provided $\gamma\neq 0$.

3.3 Radius of convergence

The nearest singularity to $z=0$ is at $z=1$, so the power series about $z=0$ converges for

$$|z|<1.$$

This is the computationally safe disk for direct series evaluation.

Practical implication

If you only ever need $z$ in $|z|<1$, you can implement HeunC yourself via the recurrence above and get robust high-precision values.
Outside $|z|<1$, you must use analytic continuation (CAS) or numerical ODE continuation.

3.4 The second local solution at $z=0$ (and logarithms)

When $\gamma\notin\mathbb{Z}$, a second independent local solution near $z=0$ has the Frobenius form

$$y_2(z)=z^{1-\gamma}\sum_{k=0}^{\infty}\tilde c_k\,z^k.$$

You can obtain it by the standard exponent-shift substitution $w(z)=z^{1-\gamma}\,v(z)$, which preserves the CHE form but shifts parameters. So (in the generic case) a local basis near $z=0$ can be taken as a local basis at $z=0$ is

$$y_1^{[0]}(z)=\mathrm{HeunC}(q,\alpha,\gamma,\delta,\epsilon;z),$$ $$y_2^{[0]}=z^{1-\gamma}\mathrm{HeunC}(q+(1-\gamma)(\epsilon-\delta),\alpha+(1-\gamma)\epsilon,2-\gamma,\delta,\epsilon;z).$$

HeunC[q, α, γ, δ, ϵ, z]
z^(1 - γ) HeunC[q + (1 - γ) (ϵ - δ), α + (1 - γ) ϵ, 2 - γ, δ, ϵ, z]

Caution

If $\gamma\in\mathbb{Z}$, the two local exponents differ by an integer and the second independent solution may contain $\log z$ terms (standard Frobenius phenomenon).

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4. Behavior near $z=1$ and the connection problem

A Frobenius basis near $z=1$ is constructed similarly with exponents $0$ and $1-\delta$ by expanding in powers of $(z-1)$. A local basis at $z=1$ is

$$y_1^{[1]}(z)=\mathrm{HeunC}(q-\alpha,-\alpha,\delta,\gamma,-\epsilon;1-z),$$ $$y_2^{[2]}(z)=(1-z)^{1-\delta}\mathrm{HeunC}(q-\alpha+(\delta-1)(\gamma+\epsilon),-\alpha+(\delta-1)\epsilon,2-\delta,\gamma,-\epsilon;1-z).$$

HeunC[q - α, -α, δ, γ, -ϵ, 1 - z]
(1 - z)^(1 - δ) HeunC[q - α + (δ - 1) (γ + ϵ), -α + (δ - 1) ϵ, 2 - δ, γ, -ϵ, 1 - z]

Most physics applications require a connection problem:

relate the local basis near $z=0$ to the local basis near $z=1$ (or to asymptotic solutions at $\infty$), and impose a boundary condition at the other end.

This is where the accessory parameter $q$ typically becomes an eigenvalue.

The local series at $z=0$ does not converge beyond $|z|=1$, but the solution does continue analytically along paths that avoid singularities and branch cuts.

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5. Asymptotic behavior at infinity (rank-1 irregular singularity)

Unlike HeunG (four regular singular points), the CHE has an irregular singularity at $z=\infty$. This means:

• solutions generally have asymptotic expansions, not convergent power series, as $|z|\to\infty$,
• different sectors in the complex plane can have different dominant behaviors (Stokes phenomenon),
• boundary conditions “at infinity” (decay/ingoing/outgoing) are subtle and often quantize parameters.

For $\epsilon\neq 0$, there are two qualitatively distinct behaviors at infinity:

• an algebraic behavior $$y(z)\sim z^{-\alpha/\epsilon}\left(1+O\!\left(\frac1z\right)\right),$$
• and an exponentially modified behavior $$y(z)\sim e^{-\epsilon z}\,z^{\alpha/\epsilon-(\gamma+\delta)} \left(1+O\!\left(\frac1z\right)\right),$$

valid in appropriate sectors of the complex plane (depending on $\arg(\epsilon z)$). Using $1/z$ as the local variable, a sectorial asymptotic basis is

$$y_1^{[\infty]}(z)=z^{-\alpha/\epsilon}\mathrm{HeunC}_\infty(q,\alpha,\gamma,\delta,\epsilon;1/z),$$ $$y_2^{[\infty]}(z)=e^{-\epsilon z}z^{\alpha/\epsilon-\gamma-\delta}\mathrm{HeunC}_\infty(q-\gamma\epsilon,\alpha-(\gamma+\delta)\epsilon,\gamma,\delta,-\epsilon;1/z).$$

z^(-α/ϵ) HeunCInf[q, α, γ, δ, ϵ, 1/z]
Exp[-ϵ z] z^(α/ϵ - γ - δ) HeunCInf[q - γ ϵ, α - (γ + δ) ϵ, γ, δ, -ϵ, 1/z]

This is the analytic backbone behind the “angular vs radial” terminology used in Ronveaux’s Part B: one often chooses the solution that matches a prescribed behavior at infinity.

Don’t ignore Stokes sectors

For irregular singularities, “the asymptotic solution” depends on direction in the complex plane.
If your physics problem selects outgoing/ingoing waves, you must align that with the correct Stokes sector for $\epsilon$ and your $z$-mapping.

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6. Special cases and reductions

6.1 The $\epsilon=0$ specialization: RCHE first, hypergeometric only if $\alpha=0$

Starting from

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

setting $\epsilon=0$ gives

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

This is the reduced confluent Heun equation (RCHE), not the Gauss equation in general. The singularity at $z=\infty$ remains irregular, although with lower rank than in the generic CHE.

Only in the further subcase $\alpha=0$ does one obtain

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

which is exactly the Gauss hypergeometric equation, with the standard identification

$$c=\gamma,\qquad a+b+1=\gamma+\delta,\qquad ab=-q.$$

So the safe rule is:

• $\epsilon=0$ with generic $\alpha$: RCHE, not hypergeometric;
• $\epsilon=0$ and $\alpha=0$: Gauss hypergeometric;
• $p\to0$ with $\alpha_R$ finite in Ronveaux: this automatically implies $(\epsilon,\alpha)\to(0,0)$, hence the hypergeometric reduction.

6.2 Mathieu, spheroidal, Coulomb spheroidal

Mathieu functions, spheroidal wave functions, and Coulomb spheroidal functions are special cases of solutions of the CHE (see DLMF §31.12 and Ronveaux Part B, §1.2).

In Ronveaux, the generalized spheroidal equation (GSE) is transformed to the non-symmetrical CHE (their Eq. (1.2.27)) by a simple linear change of variables; the resulting parameter identifications give a direct route from spheroidal problems to HeunC.

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7. Polynomial solutions (confluent Heun polynomials)

Polynomial truncations are central in quasi-exact solvability and in some boundary value problems.

There are two complementary ways to think about “polynomial solutions”, depending on which parameterization you use.

7.1 Ronveaux (1995): termination via $\alpha_R=-N$

In Ronveaux’s canonical form, the recurrence coefficient

$$h_k^{(a)} = 4p\,(k+\alpha_R-1)$$

vanishes at $k=N+1$ if

$$\alpha_R=-N,\qquad N\in\{0,1,2,\dots\}.$$

If, in addition, the recurrence produces

$$c_{N+1}^{(a)}=0,$$

then the series terminates and $Hc^{(a)}$ becomes a polynomial of degree $N$.

The condition $c_{N+1}^{(a)}=0$ is equivalent to a finite tridiagonal determinant (Ronveaux’s $\Delta_{N+1}(\sigma)=0$), yielding a polynomial equation for the accessory parameter $\sigma$ (hence for $q$ in the DLMF/Wolfram convention).

7.2 Maple/physics convention: the $(\delta_N,\Delta_{N+1})$ conditions

In the Maple convention HeunC(α,β,γ,δ,η,z) (see §8.2), polynomial truncation is often stated as the pair of conditions (Fiziev 2009):

$$\frac{\delta_M}{\alpha_M}+\frac{\beta_M+\gamma_M}{2}+N+1=0,$$

together with a second condition

$$\Delta_{N+1}(\mu)=0,$$

where $\Delta_{N+1}$ is a finite determinant (or equivalently, the vanishing of a Taylor coefficient in the three-term recurrence).

Intuitively:

• the first (“$\delta_N$”) condition kills the appropriate recurrence coefficient at step $N+1$,
• the second (“$\Delta$”) condition forces the solution into the terminating branch.

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8. Using HeunC in CAS (and not getting burned by conventions)

8.1 Wolfram Language (Mathematica)

Wolfram uses the DLMF-style parameterization:

HeunC[q, α, γ, δ, ϵ, z]

It is the solution of the CHE satisfying HeunC[..., 0] == 1, and it supports symbolic and numerical evaluation.

Useful related functions include HeunCPrime (derivative with respect to z) and series expansion via Series.

Most CAS pick a principal branch by specifying a branch cut. In Wolfram’s convention, HeunC has a branch cut from $z=1$ to $z=\infty$. If your computation crosses the cut, you must decide whether you want the value “just above” or “just below” the cut (i.e. the analytic continuation along two homotopically distinct paths).

8.2 Maple

Maple uses a different parameterization:

HeunC(alpha, beta, gamma, delta, eta, z)
HeunCPrime(alpha, beta, gamma, delta, eta, z)

A common explicit form of Maple’s CHE is

$$U''(z) +\left(\alpha_M+\frac{\beta_M+1}{z}+\frac{\gamma_M+1}{z-1}\right)U'(z) +\left(\frac{\mu}{z}+\frac{\nu}{z-1}\right)U(z)=0,$$

with

$$\mu=\frac12(\alpha_M-\beta_M-\gamma_M+\alpha_M\beta_M-\beta_M\gamma_M)-\eta_M, \qquad \nu=\frac12(\alpha_M+\beta_M+\gamma_M+\alpha_M\gamma_M+\beta_M\gamma_M)+\delta_M+\eta_M.$$

Maple’s HeunC(alpha,beta,gamma,delta,eta,z) is the local Frobenius solution around $z=0$ normalized by $U(0)=1$ (when the normalization is non-resonant).

8.3 Converting Maple parameters to Wolfram/DLMF

Given Maple parameters $(\alpha_M,\beta_M,\gamma_M,\delta_M,\eta_M)$, define $\mu,\nu$ as above. Then the equivalent DLMF/Wolfram parameters are

$$\epsilon = \alpha_M,\qquad \gamma = \beta_M+1,\qquad \delta = \gamma_M+1,$$ $$q = \mu,\qquad \alpha = \mu+\nu.$$

8.4 Converting Wolfram/DLMF parameters to Maple

Conversely, given $(q,\alpha,\gamma,\delta,\epsilon)$, set

$$\alpha_M=\epsilon,\qquad \beta_M=\gamma-1,\qquad \gamma_M=\delta-1,$$

and then

$$\delta_M=\alpha-\frac{\epsilon(\gamma+\delta)}{2},\qquad \eta_M=\frac12+\frac{\gamma(\epsilon-\delta)}{2}-q.$$

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References and further reading

• A. Ronveaux (ed.), Heun’s Differential Equations, Oxford University Press (1995), Part B (Confluent Heun equation), especially Eq. (1.2.27) and §2.2.
• NIST Digital Library of Mathematical Functions (DLMF), §31.12 “Confluent Forms of Heun’s Equation”: https://dlmf.nist.gov/31.12
• Wolfram Language Documentation: HeunC and HeunCPrime: https://reference.wolfram.com/language/ref/HeunC.html
• Maple Help: HeunC and HeunCPrime: https://www.maplesoft.com/support/help/Maple/view.aspx?path=HeunC
• P. P. Fiziev, “Novel relations and new properties of confluent Heun’s functions and their derivatives of arbitrary order” (arXiv:0904.0245): https://arxiv.org/abs/0904.0245