HeunT (Triconfluent Heun function)

What this page is (and isn’t)

The triconfluent Heun function (often denoted HeunT) is the canonical local solution of the triconfluent Heun equation (THE)—the most confluent member of the Heun family—whose only singular point is an irregular singularity at infinity. As a consequence, the standard power series about any finite point (in particular the origin) converges everywhere in $\mathbb{C}$, so HeunT is an entire function of $x$ for fixed parameters.

This page is written for researchers who use special functions in analytic/numerical work and want a reliable “how-to” reference: what equation HeunT solves, how it is normalized, how to compute it (series and asymptotics), and what special parameter regimes produce polynomials or closed forms.

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The canonical triconfluent Heun equation (THE$_1$)

A common and very convenient normalization (used in the classic reference Heun’s Differential Equations, Part E) is the canonical THE:

$$y''(x)\;-\;(\gamma+3x^2)\,y'(x)\;+\;\bigl[\alpha+(\beta-3)x\bigr]\,y(x)\;=\;0, \qquad \alpha,\beta,\gamma\in\mathbb{C}.$$

All finite points are ordinary; the only singularity is at $x=\infty$ (irregular, rank $3$).

Representative “Schrödinger” form (THE$_2$)

It is often useful (especially for asymptotics/WKB) to remove the first-derivative term. With

$$z(x)=e^{-\frac12(x^3+\gamma x)}\,y(x),$$

the canonical equation is equivalent to the representative form

$$z''(x)\;+\;\Bigl(\alpha-\frac{\gamma^2}{4}+\beta x-\frac{3}{2}\gamma x^2-\frac{9}{4}x^4\Bigr)\,z(x)=0.$$

Why two forms?

• THE$_1$ is best for defining entire solutions by initial data at $x=0$ and for power series.
• THE$_2$ is best for large-$|x|$ asymptotics (it looks like a quartic oscillator).

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Definition of HeunT (normalization)

Because $x=0$ is an ordinary point, solutions are uniquely determined by $(y(0),y'(0))$. The standard basis of entire solutions is:

• $E_1(\alpha,\beta,\gamma;x)$: the solution with $$E_1(0)=1,\qquad E_1'(0)=0;$$
• $E_2(\alpha,\beta,\gamma;x)$: the solution with $$E_2(0)=0,\qquad E_2'(0)=1.$$

In the reduced-parameter convention used on this site, the triconfluent Heun function is

$$\boxed{\;\mathrm{HeunT}(\alpha,\beta,\gamma;x)\;:=\;E_1(\alpha,\beta,\gamma;x)\;}$$

i.e. the unique entire solution of THE$_1$ with $y(0)=1$ and $y'(0)=0$.

Second solution

If you need a fundamental system, pair $\mathrm{HeunT}(\alpha,\beta,\gamma;x)$ with $E_2(\alpha,\beta,\gamma;x)$. Many CAS expose $E_2$ indirectly (e.g. via series/initial-value solving).

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Power series at the origin (global, since the function is entire)

Write

$$\mathrm{HeunT}(\alpha,\beta,\gamma;x)=\sum_{n=0}^\infty e_n(\alpha,\beta,\gamma)\,x^n, \qquad e_0=1,\;e_1=0.$$

A convenient order-3 recurrence (from the Euler-operator form of THE$_1$) is:

$$(n+2)(n+3)\,u_{n+3}-(n+2)\gamma\,u_{n+2}+\alpha\,u_{n+1}-(\beta-3-3n)\,u_n=0, \qquad n\ge 0,$$

where $u_n$ are the Taylor coefficients of a solution.

For HeunT specifically:

$$e_0=1,\qquad e_1=0,\qquad e_2=-\frac{\alpha}{2},$$

and for $n\ge 3$,

$$e_n=\frac{(n-1)\gamma\,e_{n-1}-\alpha\,e_{n-2}-(\beta+6-3n)\,e_{n-3}}{n(n-1)}.$$

The companion entire solution $E_2$

Similarly,

$$E_2(\alpha,\beta,\gamma;x)=\sum_{n=0}^\infty c'_n(\alpha,\beta,\gamma)\,x^{n+1}, \qquad c'_0=1,$$

with

$$c'_1=\frac{\gamma}{2},\qquad c'_2=\frac{\gamma^2-\alpha}{6},$$

and for $n\ge 3$,

$$c'_n=\frac{n\gamma\,c'_{n-1}-\alpha\,c'_{n-2}-(\beta+3-3n)\,c'_{n-3}}{(n+1)n}.$$

Practical series summation

Although the radius of convergence is infinite, truncated Taylor series become inefficient for very large $|x|$ because coefficients typically grow and cancellation becomes severe. For large $|x|$, use asymptotics or numerical ODE integration (see below).

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Symmetries and parameter transformations (the “$G_6$” action)

A characteristic feature of THE is a discrete symmetry under multiplication of the argument by a 6th root of unity.

Let $\omega$ satisfy $\omega^6=1$. Define the operator $T_\omega$ acting on a solution $y(\alpha,\beta,\gamma;x)$ of THE$_1$ by

$$(T_\omega y)(\alpha,\beta,\gamma;x) = \exp\!\Bigl(\tfrac12(1-\omega^3)(x^3+\gamma x)\Bigr)\; y(\omega^4\alpha,\;\omega^3\beta,\;\omega^2\gamma;\;\omega x).$$

Then:

• the six operators $T_\omega$ form a group isomorphic to the cyclic group $G_6$;
• $T_\omega$ maps solutions of THE$_1$ to solutions of THE$_1$ (with transformed parameters);
• in the THE$_2$ representation (no first derivative), $$T_\omega z(\alpha,\beta,\gamma;x)=z(\omega^4\alpha,\omega^3\beta,\omega^2\gamma;\omega x).$$

These symmetries are extremely useful for:

• generating solutions in different angular sectors,
• relating asymptotics across rotations by $\pi/3$,
• reducing computations to a “principal wedge” and rotating.

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Asymptotics at infinity and Stokes phenomenon

Since $x=\infty$ is an irregular singularity, the natural objects are formal asymptotic expansions, valid in angular sectors and related by Stokes matrices.

A formal fundamental system (THE$_1$)

A standard formal basis of solutions of THE$_1$ as $|x|\to\infty$ is:

$$\hat y_1(\alpha,\beta,\gamma;x) = x^{\frac{\beta}{3}-1}\sum_{\nu\ge 0} a_\nu(\alpha,\beta,\gamma)\,x^{-\nu},$$ $$\hat y_2(\alpha,\beta,\gamma;x) = e^{x^3+\gamma x}\,x^{-\frac{\beta}{3}-1}\sum_{\nu\ge 0}(-1)^\nu a_\nu(\alpha,-\beta,\gamma)\,x^{-\nu}.$$

The coefficients $a_\nu$ satisfy:

• normalization and first terms $$a_0=1,\qquad a_1=-\frac{\alpha}{3},\qquad a_2=\frac{1}{6}\Bigl[\gamma\Bigl(\frac{\beta}{3}-1\Bigr)+\frac{\alpha^2}{3}\Bigr],$$
• recurrence (valid for all $\nu\ge 0$) $$3(\nu+3)a_{\nu+3}+\alpha a_{\nu+2} +\gamma\Bigl(\nu+2-\frac{\beta}{3}\Bigr)a_{\nu+1} +\Bigl(\nu+1-\frac{\beta}{3}\Bigr)\Bigl(\nu+2-\frac{\beta}{3}\Bigr)a_\nu=0.$$

Stokes rays and sectors

Because the dominant exponential in $\hat y_2$ is $e^{x^3}$, the Stokes geometry is governed by $\Re(x^3)$:

• anti-Stokes directions (oscillatory/transition) satisfy $\Re(x^3)=0$, i.e. $$\arg x=\frac{\pi}{6}+k\frac{\pi}{3},\qquad k\in\mathbb{Z},$$ giving 6 rays separated by $\pi/3$;
• between these rays lie Stokes sectors where one exponential dominates.

Branching from fractional powers

The prefactors $x^{\beta/3-1}$ and $x^{-\beta/3-1}$ require a choice of branch of $\log x$ when you use these asymptotics. This is not a singularity of the function at finite $x$—it is a bookkeeping choice for asymptotic matching.

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Special parameter regimes

1) Polynomial solutions (“triconfluent Heun polynomials”)

THE$_1$ can admit polynomial solutions because it has no finite singularities: any rational solution must be a polynomial.

A polynomial solution of degree $d$ exists (and is unique up to scaling) iff both conditions hold:

$$\beta=3(d+1), \qquad \Pi_{d+1}(\alpha,\gamma)=0,$$

where $\Pi_{d+1}(\alpha,\gamma)$ is a determinantal condition (dimension $d+1$) arising from truncation of the coefficient recurrence.

A useful way to see the truncation is to substitute $P(x)=\sum_{k=0}^d \lambda_k x^k$ into THE$_1$, which yields the coefficient relations

$$(\beta-3k)\lambda_{k-1}+\alpha\lambda_k-\gamma(k+1)\lambda_{k+1}+(k+1)(k+2)\lambda_{k+2}=0,$$

valid for all integers $k$ with $\lambda_k=0$ for $k<0$ and $k>d$.

From the highest-degree conditions one gets $\beta=3(d+1)$; the remaining $d+1$ linear relations form a homogeneous system in $(\lambda_0,\dots,\lambda_d)$ whose determinant is $\Pi_{d+1}(\alpha,\gamma)$.

For small degrees one obtains explicit constraints; for instance:

• $d=0$: $\beta=3$, $\alpha=0$, with $P_0(x)=1$;
• $d=1$: $\beta=6$, $\alpha^2+3\gamma=0$, with $P_1(x)=x-\alpha/3$;
• $d=2$: $\beta=9$, $\alpha^3+12\alpha\gamma+36=0$, with an explicit quadratic polynomial.

Using this in practice

Polynomial solutions show up in quasi-exactly solvable reductions (e.g. quartic oscillators) and in spectral problems as “truncation points” for recurrence relations.

2) Liouvillian (closed-form) solutions

A Liouvillian solution is one expressible via a finite tower of algebraic operations, integration, and exponentials of integrals. For THE$_2$, a particularly concrete characterization holds:

THE$_2$ is Liouvillian iff

$$\beta\in 3\mathbb{Z}^\* \quad\text{and}\quad \Pi_{|\beta|/3}(\alpha,\gamma)=0.$$

In that case, one solution can be written as

$$P_{|\beta|/3-1}(\alpha,\gamma;x)\;e^{\frac{\varepsilon}{2}(x^3+\gamma x)}, \qquad \varepsilon=\mathrm{sign}(\beta)\in\{\pm1\},$$

with $P_m$ the corresponding polynomial from the previous section.

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How to compute HeunT in practice

If you have a CAS implementation

Most researchers should start here.

• Maple implements the reduced-parameter convention directly:

  HT := HeunT(alpha, beta, gamma, x);
  dHT := HeunTPrime(alpha, beta, gamma, x);

  Maple computes HeunT as a power series at the origin and analytically continues as needed; because the series has infinite radius of convergence, this is unusually robust for a Heun-type function.

• Wolfram Language / Mathematica provides HeunT[...] but uses a different parameterization (more parameters in the differential equation). If you want to match the 3-parameter THE$_1$ used here, convert your ODE to this canonical form first (affine change of variable plus a gauge transformation). Consult the system documentation for the exact equation it uses.

Trust but verify the convention

Before comparing two software packages (or two papers), always:

1. write down the differential equation each one uses,
2. write down the normalization conditions at the expansion point,
3. only then compare numeric values. This avoids the single most common source of “Heun mismatches.”

If you need a custom numerical implementation

A reliable strategy is hybrid:

1. Near the origin: use Taylor series with the recurrence for $e_n$.

   • Stop when the tail estimate is below your tolerance.
   • Use compensated summation if parameters produce alternating/canceling series.

2. For moderate/large $|x|$: integrate the ODE numerically along a path in the complex plane, using initial data at a point where the Taylor series is still accurate.

3. For very large $|x|$ or sector-sensitive problems (e.g. eigenvalue quantization via connection problems):

   • use the formal asymptotics $\hat y_1,\hat y_2$ within a chosen Stokes sector,
   • rotate sectors using the $G_6$ symmetry ($\omega^6=1$),
   • track Stokes rays $\arg x=\pi/6+k\pi/3$ carefully.

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Common pitfalls (and how to avoid them)

• Confusing THE$_1$ with THE$_2$: they are equivalent but not identical; mixing their normalizations will change prefactors by $e^{\pm(x^3+\gamma x)/2}$.

• Ignoring Stokes geometry: when $|x|$ is large, crossing a Stokes ray changes which asymptotic combination represents a given actual solution. If you’re matching boundary conditions at infinity, you must specify the sector.

• Resonant parameters: when $\beta/3\in\mathbb{Z}$ the fractional powers in the asymptotic prefactors simplify, but special solutions (polynomials/Liouvillian cases) may occur and numerics can become delicate.

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References (starting points)

1. A. Ronveaux (ed.), Heun’s Differential Equations, Oxford University Press (1995).
2. A. Decarreau et al., “Formes canoniques d’équations confluentes de l’équation de Heun”, Ann. Soc. Sci. Bruxelles 92 (1978) 53–78.
3. S. Yu. Slavyanov and W. Lay, Special Functions: A Unified Theory Based on Singularities, Oxford Mathematical Monographs (2000).
4. CAS documentation: Maplesoft HeunT and Wolfram Language HeunT reference pages (check the equation/normalization each uses).