Conventions and parameter maps for Heun functions

Why this page exists

The fastest way to make a wrong statement about a Heun function is to write only its name and not its equation.

For Heun functions, the symbol alone is rarely enough. In practice you must also know

1. the differential equation,
2. the parameter order,
3. the normalization or base point, and
4. the analytic-continuation convention (for example a branch cut, or a local power series around an ordinary point).

This page is the site-wide reference for those choices on heun.xyz. It is not claiming that one universal convention exists. It does the opposite: it makes our conventions explicit and gives the translations you need when moving among

• the attached reference volume Heun’s Differential Equations (Ronveaux, ed., 1995),
• the NIST Digital Library of Mathematical Functions (DLMF),
• the Wolfram Language documentation, and
• Maple’s built-in Heun functions.

The four things you should always write in a paper or notebook

For any Heun function, always record:

• the exact ODE,
• the exact normalization,
• the expansion point or base point,
• and the convention you are following (DLMF, Wolfram, Maple, Ronveaux, or your own reduced form).

Doing this once saves hours of debugging later.

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The family at a glance

Function	Singularities in a standard form	Local object used as the function	Base point on heun.xyz	Convergence / global issue
HeunG	four regular singularities $0,1,a,\infty$	Frobenius solution analytic at $0$	$z=0$	power series radius $\min(1,
HeunC	regular at $0,1$; irregular at $\infty$ of rank $1$	Frobenius solution analytic at $0$	$z=0$	power series radius $1$; principal branch cut convention matters
HeunD	two irregular singularities (standardly at $0,\infty$ or, in Maple, at $\pm1$)	local analytic solution fixed at an ordinary point	$z=1$ in the site convention	no Frobenius normalization at an irregular singular point
HeunB	regular at $0$; irregular at $\infty$ of rank $2$	Frobenius solution analytic at $0$	$z=0$	entire in $z$ for the usual local solution
HeunT	only an irregular singularity at $\infty$ of rank $3$	power-series solution fixed by two initial conditions	$z=0$	entire in $z$; Stokes behavior appears only at infinity

House style on heun.xyz

This site uses a single explicit convention on each page, but not always the same convention for all five families.

That is intentional.

• For HeunG, the site convention matches the common DLMF / Wolfram / Maple convention.
• For HeunC, the site convention matches DLMF / Wolfram and gives an explicit Maple map.
• For HeunD, the site convention follows the Wolfram/DLMF-style double-confluent equation because it keeps the irregular singularities in the canonical locations.
• For HeunB, the site convention follows the 4-parameter Ronveaux / Maple form because it is the cleanest form for local exponents, polynomial truncation, and applications.
• For HeunT, the site convention uses a reduced 3-parameter triconfluent form that is especially convenient for entire solutions and polynomial cases.

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The house conventions used on heun.xyz

HeunG on heun.xyz

We use the standard general Heun equation

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

with

$$\epsilon=\alpha+\beta-\gamma-\delta+1, \qquad a\neq 0,1.$$

Our symbol

$$\mathrm{HeunG}(a,q,\alpha,\beta,\gamma,\delta;z)$$

means the solution analytic at $z=0$ and normalized by

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

This is the easiest family conventionally because the major CAS agree on it: Maple and the Wolfram Language use the same parameter order and the same local normalization.

Practical reading rule. If a paper writes $Hl(a,q;\alpha,\beta,\gamma,\delta;z)$, that is usually the same local object as our HeunG, up to notation.

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HeunC on heun.xyz

We use the DLMF / Wolfram confluent Heun equation

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

Our symbol

$$\mathrm{HeunC}(q,\alpha,\gamma,\delta,\epsilon;z)$$

means the Frobenius solution at $z=0$ normalized by

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

The local exponents are $0$ and $1-\gamma$ at $z=0$, and $0$ and $1-\delta$ at $z=1$. The Taylor series at $z=0$ has radius of convergence $1$. In the principal Wolfram convention the branch cut runs from $z=1$ to $\infty$.

Maple-to-site map for HeunC

Maple uses

$$\mathrm{HeunC}(\alpha_M,\beta_M,\gamma_M,\delta_M,\eta_M,z)$$

for the local solution of

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

where

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

Then the equivalent site parameters are

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

Conversely, given the site parameters $(q,\alpha,\gamma,\delta,\epsilon)$,

$$\alpha_M=\epsilon, \qquad \beta_M=\gamma-1, \qquad \gamma_M=\delta-1,$$ $$\delta_M=\alpha-\frac{\epsilon(\gamma+\delta)}{2}, \qquad \eta_M=\frac12+\frac{\gamma(\epsilon-\delta)}{2}-q.$$

Good practice for HeunC

If you are comparing Maple and Wolfram values, do not compare the raw parameter tuples. Convert the tuples, substitute back into the ODE, and only then compare function values.

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HeunD on heun.xyz

We use the Wolfram/DLMF-style double-confluent equation

$$z^2y''(z)+\bigl(\gamma+\delta z+\epsilon z^2\bigr)y'(z)+(\alpha z-q)y(z)=0,$$

or equivalently

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

Our symbol

$$\mathrm{HeunD}(q,\alpha,\gamma,\delta,\epsilon;z)$$

means the local analytic solution fixed at the ordinary point $z=1$ by

$$y(1)=1, \qquad y'(1)=0.$$

This choice is not cosmetic. In this convention $z=0$ is an irregular singular point, so there is no Frobenius normalization there.

Relation to the DLMF reduced DCHE

DLMF writes the doubly confluent equation in the reduced form

$$y''(z)+\left(\frac{\delta_D}{z^2}+\frac{\gamma_D}{z}+1\right)y'(z) +\frac{\alpha_D z-q_D}{z^2}\,y(z)=0.$$

If $\epsilon\neq 0$, a rescaling of the independent variable reduces the site convention to the DLMF normalization with the constant term in $y'$ fixed to $1$. After that rescaling, the two conventions differ only by a relabeling of the coefficients of $1/z^2$ and $1/z$.

Maple warning for HeunD

Maple also has a built-in HeunD, but it uses a different normal form: the irregular singularities are placed at $x=\pm1$, the origin is an ordinary point, and the built-in function is expanded around $x=0$. So Maple HeunD is not a simple parameter relabeling of the site convention.

Translation rule. For HeunD, never compare software output by symbol name alone. Compare by ODE and normalization.

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HeunB on heun.xyz

We use the 4-parameter Ronveaux / Maple biconfluent form

$$z\,y''(z)+(1+\alpha-\beta z-2z^2)y'(z) +\Bigl((\gamma-\alpha-2)z-\tfrac12\bigl[\delta+(1+\alpha)\beta\bigr]\Bigr)y(z)=0.$$

Our symbol

$$\mathrm{HeunB}(\alpha,\beta,\gamma,\delta;z)$$

means the solution analytic at $z=0$ and normalized by

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

This convention is extremely convenient because

• the exponents at $z=0$ are transparent,
• the local power series is straightforward,
• polynomial truncation is easy to state,
• and Maple uses the same 4-parameter structure.

Exact map from the site convention to Wolfram HeunB

Wolfram uses the 5-parameter equation

$$z\,y''(z)+\bigl(\gamma_W+\delta_W z+\epsilon_W z^2\bigr)y'(z)+(\alpha_W z-q_W)y(z)=0,$$

with the regular solution normalized by $y(0)=1$.

The exact parameter map from the site convention to Wolfram’s parameters is

$$\epsilon_W=-2, \qquad \gamma_W=1+\alpha, \qquad \delta_W=-\beta,$$ $$\alpha_W=\gamma-\alpha-2, \qquad q_W=\frac12\bigl(\delta+(1+\alpha)\beta\bigr).$$

DLMF warning for HeunB

DLMF uses a different reduced biconfluent form,

$$y''(z)-\left(\frac{\gamma_D}{z}+\delta_D+z\right)y'(z) +\frac{\alpha_D z-q_D}{z}\,y(z)=0,$$

which is equivalent to the site convention only after a rescaling of the independent variable and a linear redefinition of parameters.

So for HeunB the correct question is not “which symbol is right?”, but “which ODE am I actually solving?”

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HeunT on heun.xyz

We use a reduced 3-parameter triconfluent form,

$$y''(z)-\bigl(\gamma+3z^2\bigr)y'(z)+\bigl[\alpha+(\beta-3)z\bigr]y(z)=0.$$

Our symbol

$$\mathrm{HeunT}(\alpha,\beta,\gamma;z)$$

means the entire solution fixed by

$$y(0)=1, \qquad y'(0)=0.$$

This convention is well adapted to entire-function methods, Taylor recurrences, and polynomial solutions.

Relation to Wolfram HeunT

Wolfram uses the 5-parameter triconfluent equation

$$y''(z)+\bigl(\gamma_W+\delta_W z+\epsilon_W z^2\bigr)y'(z) +(\alpha_W z-q_W)y(z)=0,$$

with the power-series normalization

$$y(0)=1, \qquad y'(0)=0.$$

If you want the site convention with the same independent variable and no further gauge transformation, then it is the specialization

$$\gamma_W=-\gamma, \qquad \delta_W=0, \qquad \epsilon_W=-3,$$ $$\alpha_W=\beta-3, \qquad q_W=-\alpha.$$

DLMF warning for HeunT

DLMF uses another 3-parameter triconfluent standard form,

$$y''(z)+(\gamma_D+z)z\,y'(z)+(\alpha_D z-q_D)y(z)=0,$$

which belongs to the same triconfluent family but is not the same literal ODE as the site convention. In the triconfluent literature, different normal forms are common, and one often passes among them by normal-form reductions, rescalings, and gauge transformations.

So for HeunT, more than for any other member of the family, you should compare by the ODE itself and not by the function name alone.

About Maple HeunT

Maple provides a 3-parameter HeunT defined by a power series at the origin. Its interface is much closer in spirit to the reduced convention used on this site than to Wolfram’s 5-parameter form, but you should still check the exact defining ODE before transferring identities or special values.

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A compact crosswalk of the five site conventions

Site page	House symbol on heun.xyz	Closest direct CAS match	What to watch for
General	HeunG(a,q,\alpha,\beta,\gamma,\delta;z)	Maple and Wolfram both match directly	remember the Fuchs relation for $\epsilon$
Confluent	HeunC(q,\alpha,\gamma,\delta,\epsilon;z)	Wolfram matches directly	Maple uses $(\alpha_M,\beta_M,\gamma_M,\delta_M,\eta_M)$ and a different ODE presentation
Double confluent	HeunD(q,\alpha,\gamma,\delta,\epsilon;z)	Wolfram matches directly	Maple HeunD uses a different canonical variable placement and a different base point
Biconfluent	HeunB(\alpha,\beta,\gamma,\delta;z)	Maple is the closest direct match	Wolfram uses a 5-parameter form; DLMF uses a reduced form
Triconfluent	HeunT(\alpha,\beta,\gamma;z)	reduced 3-parameter literature / Maple-style workflows	Wolfram uses a 5-parameter form; DLMF uses another reduced normal form

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Which parameter is the accessory parameter?

This matters in spectral problems.

• For HeunG, the accessory parameter is $q$.
• For HeunC, the accessory parameter is again $q$ in the DLMF / Wolfram convention; Maple packages the same freedom into $\eta_M$ together with the other coefficients.
• For HeunD, the accessory parameter is naturally $q$ in the Wolfram / DLMF-like convention.
• For HeunB, in the 4-parameter Ronveaux / Maple form the role most often played by an accessory parameter is carried by $\delta$.
• For HeunT, different normal forms distribute the free coefficients differently; in the reduced 3-parameter form used on this site there is no universally preferred “one true accessory parameter” in the same way as for HeunG or HeunC.

Practical rule. The accessory parameter is the one not fixed by the local exponent data and usually determined by a global boundary condition.

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Polynomial conditions: the site-wide rule of thumb

The detailed truncation formulas depend strongly on the chosen convention, but the overall pattern is stable.

• HeunG: one exponent parameter must be a nonpositive integer, and the accessory parameter must satisfy a finite algebraic condition.
• HeunC: a linear truncation condition plus a determinant or accessory-root condition.
• HeunB: a linear truncation condition plus a root condition in the accessory-like parameter.
• HeunD: the usual double-confluent normalizations do not lead to the same simple Frobenius-polynomial story, because the canonical equation has no regular singular point at which a polynomial truncation is naturally defined.
• HeunT: polynomial solutions occur when one coefficient parameter takes a positive-integer value and the remaining parameters satisfy a finite determinant condition.

So when you quote a “Heun polynomial condition,” always quote it in the same convention as the ODE you are using.

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The common traps this site is designed to eliminate

1. Confusing a local solution with a global eigenfunction

HeunG, HeunC, HeunB, and the ordinary-point normalizations of HeunD and HeunT are first and foremost local normalized solutions. In applications you often want something else: a solution satisfying conditions at multiple singular points or in multiple Stokes sectors. That usually turns the accessory parameter into an eigenvalue.

2. Comparing Maple and Wolfram values without converting parameters

This is harmless for HeunG, but dangerous for HeunC, very dangerous for HeunD, and often fatal for HeunB and HeunT.

3. Forgetting that irregular singularities change what “normalization” means

At a regular singular point you can define a Frobenius solution. At an irregular singular point you usually cannot. Then the function is defined instead by initial data at an ordinary point or by an asymptotic condition in a Stokes sector.

4. Treating branch issues and Stokes issues as if they were the same thing

• HeunG and HeunC have branch-cut conventions for principal values.
• HeunD, HeunB, and HeunT are better thought of in terms of local analytic continuation and Stokes phenomena near irregular singular points.

5. Switching conventions in the middle of a derivation

Pick one convention, do the algebra in that convention, and only translate at the beginning or at the end.

Best workflow

When you inherit an equation from a paper, do the following in order:

1. identify the singularity pattern,
2. rewrite the ODE in one chosen canonical form,
3. record the normalization,
4. convert parameters once,
5. and only then call a CAS or compare formulas.

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What every function page on heun.xyz should state up front

Every page on this site should begin with a compact definition box containing the following items.

1. Exact ODE. No shorthand, no hidden parameter constraints.
2. Normalization. For example $y(0)=1$ or $y(1)=1$, $y'(1)=0$.
3. Singularity structure. Regular vs irregular, and rank when irregular.
4. Expansion point. The point around which the local series is defined.
5. Closest CAS match. Maple, Wolfram, both, or neither directly.
6. Parameter map note. A one-line warning if the same symbol means something else elsewhere.

That single box already removes most avoidable confusion.

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References and authoritative sources

Core mathematical references

• A. Ronveaux (ed.), Heun’s Differential Equations, Oxford University Press, 1995. In particular Part B (confluent), Part C (double confluent), Part D (biconfluent), and Part E (triconfluent).
• NIST Digital Library of Mathematical Functions, Chapter 31, especially §31.2 and §31.12: https://dlmf.nist.gov/31 and https://dlmf.nist.gov/31.12

Official CAS documentation

• Wolfram Language: HeunG, HeunC, HeunD, HeunB, HeunT.
  • https://reference.wolfram.com/language/ref/HeunG.html
  • https://reference.wolfram.com/language/ref/HeunC.html
  • https://reference.wolfram.com/language/ref/HeunD.html
  • https://reference.wolfram.com/language/ref/HeunB.html
  • https://reference.wolfram.com/language/ref/HeunT.html
• Maple Help: HeunG, HeunC, HeunD, HeunB, HeunT.
  • https://www.maplesoft.com/support/help/Maple/view.aspx?path=HeunG
  • https://www.maplesoft.com/support/help/Maple/view.aspx?path=HeunC
  • https://www.maplesoft.com/support/help/Maple/view.aspx?path=HeunD
  • https://www.maplesoft.com/support/help/Maple/view.aspx?path=HeunB
  • https://www.maplesoft.com/support/help/Maple/view.aspx?path=HeunT

Suggested use of this page

Use this page first, then go to the function-specific guide:

• HeunG for the general equation,
• HeunC for the singly confluent case,
• HeunD for the double-confluent case,
• HeunB for the biconfluent case,
• HeunT for the triconfluent case.

When those pages mention a “site convention,” they mean exactly the conventions fixed here.