HeunB (Biconfluent Heun Function)

What this page is (and is not)

This is a researcher-facing guide to the biconfluent Heun function HeunB, i.e. the local solution at the regular singular point $z=0$ of the biconfluent Heun equation (BHE). The BHE is one of the four confluent descendants of the general Heun equation and is characterized by

• one regular singularity (at $z=0$ in standard forms), and
• one irregular singularity of rank 2 (at $z=\infty$), hence Stokes phenomena at infinity.

This page aims to help you

• recognize when your ODE is (or can be reduced to) the BHE,
• navigate competing canonical forms and parameter conventions (Ronveaux/Maroni, Maple, Wolfram, DLMF),
• understand what the function HeunB actually means (normalization, analytic continuation, branches),
• compute HeunB reliably (series, ODE integration, asymptotics) and validate results,
• use truncation / polynomial cases that frequently appear in quantum-mechanical eigenvalue problems.

It does not attempt to reproduce the full theory of the BHE (monodromy/Stokes matrices, global connection problems, etc.). Instead, it emphasizes what you need to use HeunB correctly in actual research work.

Always write the ODE and the normalization

“HeunB” is not a single universal object. Different sources fix different canonical equations (and even different numbers/names of parameters).
Always record (i) the exact ODE you mean and (ii) the normalization condition at $z=0$.

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1. The canonical biconfluent Heun equation behind HeunB

1.1 Ronveaux/Maroni canonical form (4 parameters)

The canonical BHE used in Ronveaux (1995, Part D) can be written as (Maroni’s Eq. (1.2.5))

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

Dividing by $z$ makes the singularity structure explicit:

$$y''(z) + \left(\frac{1+\alpha}{z}-\beta-2z\right) y'(z) + \left(\gamma-\alpha-2 - \frac{\delta + (1+\alpha)\beta}{2z}\right) y(z)=0.$$

• $z=0$ is a regular singular point.
• $z=\infty$ is an irregular singular point of rank 2 (the $-2z\,y'$ term drives the Stokes behavior).

This 4-parameter form is the one most closely aligned with Maple’s HeunB(α,β,γ,δ,z) and with much of the mathematical literature following Ronveaux.

1.2 The normal (Schrödinger-like) form

Removing the first-derivative term by

$$u(z)=z^{\frac{1+\alpha}{2}} e^{-\frac{\beta}{2}z-\frac12 z^2}\,y(z)$$

transforms the BHE into a “normal form” (Maroni’s Eq. (1.3.1)):

$$u''(z) + \left[ \frac{1-\alpha^2}{4z^2} - \frac{\delta}{2z} + \gamma - \left(\frac{\beta}{2}\right)^2 - \beta z - z^2 \right] u(z)=0.$$

This is extremely useful if you meet the BHE via a radial Schrödinger equation: the bracket looks like an effective potential with $z^2$ and $z$ terms plus centrifugal/Coulomb-type pieces.

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2. Definition of the biconfluent Heun function HeunB

2.1 Local exponents at the origin

A Frobenius ansatz $y(z)=\sum_{n\ge 0} c_n z^{n+\rho}$ gives the indicial equation

$$\rho(\rho+\alpha)=0,$$

so the characteristic exponents at $z=0$ are

$$\rho_1=0,\qquad \rho_2=-\alpha.$$

• If $\alpha\notin\mathbb{Z}$, the two local solutions are (generically) a regular one and a singular one $\sim z^{-\alpha}$.
• If $\alpha\in\mathbb{Z}$, logarithms can appear in the second solution (see §4.2).

2.2 The HeunB/Ronveaux normalization

The biconfluent Heun function HeunB is the solution that is analytic at $z=0$ and normalized by

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

In Ronveaux/Maroni notation this solution is denoted $N(\alpha,\beta,\gamma,\delta;z)$ and is an entire function of $z$ (no other finite singularities exist).

Entire in $z$ does not mean “numerically easy”

Because the only singularities are at $0$ and $\infty$, the analytic solution at $0$ analytically continues to an entire function.
But numerically, growth like $e^{z^2}$ in some sectors can cause overflow/instability; see §6.

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3. Power-series expansion at $z=0$ (the workhorse)

3.1 Series form used by Ronveaux/Maroni (and Maple)

Write the analytic solution as (Maroni’s Eq. (3.1.3))

$$\mathrm{HeunB}(\alpha,\beta,\gamma,\delta;z) =\sum_{n=0}^{\infty}\frac{A_n}{(1+\alpha)_n\,n!}\,z^n, \qquad (1+\alpha)_n=\frac{\Gamma(1+\alpha+n)}{\Gamma(1+\alpha)}.$$

The first coefficients are (Eqs. (3.1.4)–(3.1.5))

$$A_0=1,\qquad A_1=\tfrac12\bigl(\delta+\beta(1+\alpha)\bigr),$$

and for $n\ge 0$,

$$\boxed{ A_{n+2} =\left((n+1)\beta+\tfrac12(\delta+\beta(1+\alpha))\right)A_{n+1} -(n+1)(n+1+\alpha)\,(\gamma-2-\alpha-2n)\,A_n. }$$

This recurrence is stable and easy to implement for moderate $n$, and it is the fastest way to evaluate HeunB when $|z|$ is not too large.

3.2 Initial derivative at the origin

From the series one immediately reads

$$\mathrm{HeunB}'(\alpha,\beta,\gamma,\delta;0)=\frac{A_1}{1+\alpha} =\frac{\delta+\beta(1+\alpha)}{2(1+\alpha)} =\frac{\beta}{2}+\frac{\delta}{2(1+\alpha)}.$$

This is the correct initial slope for ODE integration starting at $z=0$ (or at a small $z=z_0$).

3.3 A practical series-evaluation recipe

For complex parameters, evaluate

$$S_N(z)=\sum_{n=0}^{N}\frac{A_n}{(1+\alpha)_n\,n!}\,z^n$$

with the recurrence above and stop when the last term is below your target tolerance.

Implementation notes:

• Use logarithmic updates for $(1+\alpha)_n$ and $n!$ if you need very high $N$.
• For high precision, compute with arbitrary precision arithmetic (e.g. mpmath, Mathematica, Maple).
• If $|z|$ is large, the power series may need many terms; then ODE integration or asymptotics is usually better (§6).

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4. The second local solution and singular parameter cases

4.1 Generic case $\alpha\notin\mathbb{Z}$

When $\alpha$ is not an integer, a convenient fundamental system at the origin is (Maroni’s Prop. 3.1.1)

$$y_1(z)=\mathrm{HeunB}(\alpha,\beta,\gamma,\delta;z),\qquad y_2(z)=z^{-\alpha}\,\mathrm{HeunB}(-\alpha,\beta,\gamma,\delta;z).$$

The Wronskian of any two solutions $y_1,y_2$ of the BHE satisfies

$$W[y_1,y_2](z)=C\,z^{-(1+\alpha)} e^{\beta z + z^2},$$

and in particular for the pair above $C=-\alpha$.

4.2 Relative-integer cases and logarithms

If $\alpha\in\mathbb{Z}$, the exponent difference $\rho_1-\rho_2=\alpha$ is an integer, and the second solution may contain a $\log z$ term. This is not special to Heun; it is the standard Frobenius phenomenon.

Practical implication: if you need the second solution and your parameters place you in a resonant case, do not “guess” it from $z^{-\alpha}\mathrm{HeunB}(-\alpha,\dots)$ without checking whether the logarithmic term is present.

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5. Polynomial (terminating) cases: “HeunB polynomials”

A major reason HeunB appears in physics is that bound-state/eigenvalue conditions often force the analytic solution to truncate into a polynomial times a simple gauge factor.

From the recurrence for $A_n$, the analytic solution becomes a polynomial of degree $m$ if and only if (Maroni §3.3)

$$\gamma-\alpha-2=2m,\qquad A_{m+1}=0.$$

• The first condition fixes $\gamma$ relative to $\alpha$ (think “quantization of an exponent at infinity”).
• The second condition is an algebraic constraint on $\delta$ (for fixed $\alpha,\beta$), typically solved as a root condition.

More precisely, for fixed $(\alpha,\beta)$ and integer $m\ge 0$, the coefficient $A_{m+1}$ is a polynomial of degree $m+1$ in $\delta$, so there are at most $m+1$ values $\delta=\delta_\mu^{(m)}$ that produce truncation.

5.1 Orthogonality (when parameters are real)

When $1+\alpha>0$ and $\beta\in\mathbb{R}$, the resulting polynomial solutions satisfy an orthogonality relation (Maroni Eq. (3.3.4)):

$$\int_0^\infty t^\alpha e^{-\beta t-t^2}\, P_{m,\mu}(t)\,P_{m,\nu}(t)\,dt=0, \qquad \mu\ne \nu,$$

with $P_{m,\mu}$ the $m$-th degree polynomial solution.

This is one route from HeunB to families of orthogonal polynomials that arise in quasi-exactly solvable models.

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6. Behavior at infinity and why Stokes sectors matter

The irregular singularity at $z=\infty$ is where most “hard” BHE problems live (connection problems, eigenvalues, Stokes multipliers).

Maroni constructs canonical solutions in different half-planes. The key takeaway is that, as $|z|\to\infty$ in a suitable sector, there is a pair of asymptotic behaviors of the form (see §3.4):

$$B^+(z)\sim z^{\frac{\gamma-\alpha-2}{2}}\sum_{n=0}^\infty \frac{a_n}{z^n},$$

and

$$H^+(z)\sim z^{-\frac{\gamma+\alpha+2}{2}}\,e^{\beta z+z^2}\sum_{n=0}^\infty \frac{e_n}{z^n}.$$

So generically, one solution is algebraic and the other carries the dominant exponential $e^{z^2+\beta z}$.

Eigenvalue problems are “kill the growing exponential” problems

In many physics applications you want a solution that decays (or is subdominant) along a chosen ray (e.g. $z\in\mathbb{R}_+$).
That typically means: take the analytic-at-0 solution, analytically continue it, and impose that its coefficient of the $e^{z^2+\beta z}$ piece vanishes.
This becomes a nonlinear condition on parameters (often the accessory parameter $\delta$).

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7. Special cases and reductions (useful checks)

7.1 Reduction to confluent hypergeometric ${}_1F_1$

A particularly important specialization is (Maroni Eq. (3.1.12)):

$$\mathrm{HeunB}(\alpha,0,\gamma,0;z) ={}_1F_1\!\left(\frac{\alpha+2-\gamma}{4}\,;\,1+\alpha\,;\,\frac{z^2}{2}\right).$$

This identity is a powerful validation tool: if you set $\beta=\delta=0$ and compare to a high-quality ${}_1F_1$ implementation, your code for HeunB should match.

7.2 Elementary exponential solution

Another exact identity (Maroni Eq. (3.1.16)) is

$$\mathrm{HeunB}\bigl(\alpha,\beta,-(\alpha+2),\beta(1+\alpha);z\bigr)=e^{\beta z+z^2}.$$

Again, this is excellent for sanity checks.

7.3 Laguerre-polynomial specializations

When parameters enforce polynomial truncation with $\beta=0$, the polynomial solutions reduce to generalized Laguerre polynomials $L_m^{(\alpha)}$ (Maroni Eq. (3.3.5)). If your application predicts a Laguerre limit, verify that your HeunB reduces accordingly.

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7.4 Symmetries and functional relations (quick consistency checks)

Beyond special-case reductions, the BHE has nontrivial discrete transformations that map solutions to solutions while reshuffling parameters. In Ronveaux/Maroni notation (Prop. 3.1.2), for $\alpha\notin\{-1,-2,\dots\}$ one has

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

and a “$\pi/2$ rotation” identity

$$\mathrm{HeunB}(\alpha,\beta,\gamma,\delta;z) =e^{\beta z+z^2}\,\mathrm{HeunB}(\alpha,-i\beta,-\gamma,i\delta;-iz).$$

These identities are useful in practice for:

• sanity checks (numerical evaluation must satisfy them),
• moving from a numerically unstable ray to a more stable one (rotate $z$),
• relating Stokes sectors at infinity.

Do not confuse identities with connection formulas

The relations above are exact symmetries (coming from explicit transformations of the differential equation).
They do not solve the hard global problem of expressing the origin-normalized solution in a basis adapted to a specific Stokes sector at infinity. That is a genuine connection/Stokes problem.

8. Conventions across references (Ronveaux/Maple vs Wolfram vs DLMF)

8.1 Maple’s HeunB(α,β,γ,δ,z)

Maple defines HeunB(α,β,γ,δ,z) as the Frobenius solution about $z=0$ for a 4-parameter canonical BHE, normalized by HeunB(...,0)=1, and computed as a power series (hence convergent for all $z\in\mathbb{C}$).

• Maple help page: HeunB — The Heun Biconfluent function.

8.2 Wolfram Language’s HeunB[q,α,γ,δ,ϵ,z]

Wolfram uses a 5-parameter bi-confluent equation (as displayed in its documentation):

$$z\,y''(z) + (\gamma + \delta z + \epsilon z^2)\,y'(z) + (\alpha z - q)\,y(z)=0, \qquad y(0)=1.$$

To match Ronveaux/Maple parameters $(\alpha,\beta,\gamma,\delta)$ to Wolfram parameters $(q,\alpha,\gamma,\delta,\epsilon)$, set

$$\epsilon=-2,\qquad \gamma_{\mathrm{W}}=1+\alpha,\qquad \delta_{\mathrm{W}}=-\beta,\qquad \alpha_{\mathrm{W}}=\gamma-\alpha-2,\qquad q_{\mathrm{W}}=\tfrac12\bigl(\delta+(1+\alpha)\beta\bigr),$$

and use the same $z$.

(Here the subscript “W” means “Wolfram parameter”.)

8.3 DLMF’s biconfluent equation

DLMF §31.12.3 uses a different “standard form”:

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

This is equivalent to the Ronveaux/Maple form by a rescaling of $z$ and a linear redefinition of parameters. If you are translating a paper that quotes DLMF parameters, do the conversion once and then stay in a single convention throughout your work.

Pick one convention and stick to it

For multi-step derivations (physics reductions, connection problems, numerics), switching conventions mid-stream is a common source of sign and factor-of-two errors.

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9. Reliable computation workflows

9.1 In CAS (Maple / Mathematica)

Recommended workflow:

1. Write down the ODE you believe you have.
2. Reduce it to your chosen canonical BHE by explicit substitutions (affine change of $z$, gauge factor $e^{az+bz^2}$, power prefactor $z^\rho$, etc.).
3. Translate parameters into your CAS convention (Maple vs Wolfram).
4. Validate with:
   • series at $z=0$,
   • a special-case reduction (§7),
   • numerical residual of the ODE (substitute and check it solves to tolerance).

9.2 In your own code (series + ODE integration)

A robust pattern is:

• Use the power series to initialize at a small $z=z_0$ (not exactly $0$ if your ODE solver dislikes singular coefficients).
• Integrate the ODE along a path in the complex plane using an adaptive solver.
• If you need large-$|z|$ behavior, compare to the asymptotic forms (§6) in the correct sector.

Below is a minimal series evaluator following Maroni’s recurrence:

import mpmath as mp

def heunB_series(alpha, beta, gamma, delta, z, nterms=50):
    # A_n recurrence (Maroni 1995, Eq. 3.1.5)
    A0 = mp.mpf(1)
    A1 = mp.mpf('0.5')*(delta + beta*(1+alpha))
    A = [A0, A1]
    for n in range(0, nterms-2):
        A_np2 = ((n+1)*beta + mp.mpf('0.5')*(delta + beta*(1+alpha)))*A[n+1] \
                - (n+1)*(n+1+alpha)*(gamma-2-alpha-2*n)*A[n]
        A.append(A_np2)

    # sum A_n / ((1+alpha)_n n!) z^n
    s = mp.mpf(0)
    poch = mp.mpf(1)  # (1+alpha)_0
    fact = mp.mpf(1)  # 0!
    for n, An in enumerate(A):
        if n > 0:
            poch *= (alpha + n)  # (1+alpha)_n = ∏_{k=1}^n (alpha + k)
            fact *= n
        s += An/(poch*fact) * (z**n)
    return s

Be careful near $\alpha=-1,-2,\dots$

The representation with $(1+\alpha)_n$ has poles when $\alpha$ is a negative integer. The underlying solution can still exist (often by a limiting procedure), but you must treat these cases separately.

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10. A Laplace-type integral relation (advanced but useful)

Maroni derives integral transforms that connect the origin-normalized solution $N(\alpha,\beta,\gamma,\delta;z)$ to canonical solutions at infinity.

One representative Laplace-type relation (see §4.1) is:

$$B^+(\hat{\alpha},\hat{\beta},\hat{\gamma},\hat{\delta};z) =\frac{2^{1+\alpha}}{\Gamma(1+\alpha)} \int_0^\infty e^{-2zx}\,x^\alpha e^{-\beta x-x^2}\, \mathrm{HeunB}(\alpha,\beta,\gamma,\delta;x)\,dx,$$

valid (at least) for $\Re z>0$ and $\Re\alpha>-1$, with

$$\hat{\alpha}=\tfrac12(\alpha-\gamma),\qquad \hat{\beta}=\beta,\qquad \hat{\gamma}=-\tfrac12(\gamma+3\alpha),\qquad \hat{\delta}=\delta+\tfrac12\beta(\gamma+\alpha).$$

You do not need this formula for day-to-day evaluation of HeunB, but it is valuable for:

• proving properties (growth, analytic continuation),
• building alternative numerical schemes (quadrature-based),
• understanding how “origin data” and “infinity data” are related.

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References and recommended primary sources

• P. P. Maroni, “The Biconfluent Heun Equation”, in A. Ronveaux (ed.), Heun’s Differential Equations, Oxford University Press (1995), Part D.
• NIST Digital Library of Mathematical Functions (DLMF), Chapter 31 (Heun Functions), especially §31.12 (confluent forms) and §31.18 (computation).
• Maple help: HeunB — The Heun Biconfluent function.
• Wolfram Language documentation: HeunB.