Introduction

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Heun functions are solutions of the second-order Fuchsian differential equation with four regular singular points, extending the hypergeometric case by introducing an accessory parameter $q$ that is not determined by the local exponents. When singularities coalesce, one obtains the confluent, doubly confluent, biconfluent, and triconfluent forms (HeunC, HeunD, HeunB, HeunT), which have irregular singularities. Heun equations emerge at the crossroads of mathematical physics, including isomonodromic deformation (which preserves monodromy as singularities move, leading to the Painlevé equations), the ODE/IM correspondence, resurgence, and 2D CFTs. Heun equations have wide applications in modern physics, including black hole perturbations, AdS/CFT duality, and quantum mechanics. Hypergeometric ruled the 20th century—Heun is shaping the 21st.

1.  HeunG (General Heun Equation)

[0, 1, $t$, $\infty$]: It has four regular singularities at 0, 1, $t$, and $\infty$. The Heun equation is the most general second-order linear ODE with four regular singularities. The standard form is

$$y''(z)+\left(\frac{\gamma}{z}+\frac{\delta}{z-1}+\frac{\epsilon}{z-t}\right)y'(z)+\frac{\alpha\beta z-q}{z(z-1)(z-t)}y(z)=0$$

where $\epsilon=\alpha+\beta-\gamma-\delta+1$, and $q$ is the accessory parameter. A local Heun funtion $H\ell(t,q;\alpha,\beta,\gamma,\delta;z)$ is a Taylor series solution at $z=0$. In the books [1, 2], a “Heun function” is defined as a local Heun function that is also regular at $z=1$, which gives an eigenvalue problem. We prefer the convention that the term “Heun function” is equivalent to the local Heun function, not an eigenfunction. (As a comparison, the hypergeometric function is also a local solution.) This is exactly the HeunG implemented by Wolfram and Maple.

If we impose boundary conditions at both $z=0$ and $z=1$, we have an eigenvalue problem. A Heun polynomial is a solution regular at $z=0$, $1$, and $t$, which is beyond an eigenvalue problem.

HeunG in Wolfram, HeunG in Maple, HeunG in DLMF

2.  HeunC (Confluent Heun Equation)

$[0,\,1,\,\infty^{(1)}]$: It has two regular singularities at 0 and 1, and an irregular singularity of rank 1 at $\infty$. $$ y''(z)+\left(\frac{\gamma}{z}+\frac{\delta}{z-1}+\epsilon\right)y'(z)+\frac{\alpha z-q}{z(z-1)}y(z)=0 $$

A local solution at $z=0$ is $\text{HeunC}(q,\alpha ,\gamma ,\delta ,\epsilon;z)$. A solution at the irregular singularity $z=\infty$ is $z^{-\alpha/\epsilon}\,\text{HeunC}_\infty(q,\alpha ,\gamma ,\delta ,\epsilon; z^{-1})$, where $\text{HeunC}_\infty$ is an asymptotic series.

HeunC in Wolfram, HeunC in DLMF

HeunC in Maple uses another form

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

HeunRC (Reduced Confluent Heun Equation)

3. HeunD (Doubly Confluent Heun Equation)

$[0^{(1)},\,\infty^{(1)}]$: It has two irregular singularities of rank 1 at 0 and $\infty$. $$ y''+\left(\frac{\gamma}{z^{2}}+\frac{\delta}{z}+\epsilon\right)y'+\frac{\alpha z-q}{z^{2}}y=0 $$

HeunD in Wolfram

The symmetric canonical form is $$ D^2 y+ \alpha \left( z + \frac{1}{z} \right) D y+\left((\beta_1+\frac12)\alpha z+(\frac{\alpha^2}{2}-\gamma)+(\beta_{-1}-\frac12)\frac{\alpha}{z}\right)y(z)=0, {\qquad\text{where } D\equiv z\frac{d}{dz}} $$ The second and first derivative part is $z^2y''(z)+(z+\alpha z^2+\alpha)y'(z)$. The symmetric canonical form is useful to make connections to the Mathieu equation.

$$\llap{\text{DLMF:}\qquad} \frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+\left(\frac{\delta}{z^{2}}+\frac{\gamma}{z}+1\right)\frac{\mathrm{d}w}{\mathrm{d}z}+\frac{\alpha z-q}{z^{2}}w=0.$$

HeunRD (Reduced Doubly Confluent Heun Equation)

HeunDD (Doubly Reduced Doubly Confluent Heun Equation)

4. HeunB (Biconfluent Heun Equation)

$[0,\,\infty^{(2)}]$: It has a regular singularity at 0, and an irregular singularity of rank 2 at $\infty$. $$ y''+\left(\frac{\gamma}{z}+\delta+\epsilon z\right)y'+\frac{\alpha z-q}{z}y=0 $$

HeunB in Wolfram

$$\llap{\text{DLMF:}\qquad}\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}-\left(\frac{\gamma}{z}+\delta+z\right)\frac{\mathrm{d}w}{\mathrm{d}z}+\frac{\alpha z-q}{z}w=0.$$

HeunRB (Reduced Biconfluent Heun Equation)

5. HeunT (Triconfluent Heun Equation)

$[\infty^{(3)}]$: It has an irregular singularity of rank 3 at $\infty$. $$ y''+\left(\gamma+\delta z+\epsilon z^2\right)y'+(\alpha z-q)y=0 $$

HeunT in Wolfram

$$\llap{\text{DLMF:}\qquad} \frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+\left(\gamma+z\right)z\frac{\mathrm{d}w}{\mathrm{d}z}+\left(\alpha z-q\right)w=0.$$

HeunRT (Reduced Triconfluent Heun Equation)

Mathieu equation

The standard form of the Mathieu equation is $$ y^{\prime\prime}+(a-2q\cos\left(2z\right))y=0 $$ The modified Mathieu equation ($z\to iz$, $\bar{a}\equiv -a$, $\bar{q}\equiv -q$) is $$ -y^{\prime\prime}+2\bar{q}\cosh\left(2z\right)y=\bar{a}y $$

Mathieu in Wolfram, Mathieu in Maple, Mathieu in DLMF

Resources

Books

[1] A. Ronveaux (Ed.), Heun's Differential Equations (Oxford University Press, Oxford; New York, 1995).

[2] S. Yu. Slavyanov and W. Lay, Special Functions: A Unified Theory Based on Singularities (Oxford University Press, Oxford; New York, 2000).

Papers

K. Heun, Zur Theorie der Riemann'schen Functionen zweiter Ordnung mit vier Verzweigungspunkten, Math. Ann. 33, 161 (1888).

R. S. Maier, The 192 solutions of the Heun equation, Math. Comp. 76, 811 (2006) [arXiv:math/0408317].

O. Lisovyy and A. Naidiuk, Accessory parameters in confluent Heun equations and classical irregular conformal blocks, Lett Math Phys 111, 137 (2021) [arXiv:2101.05715].

O. Lisovyy and A. Naidiuk, Perturbative connection formulas for Heun equations, J. Phys. A: Math. Theor. 55, 434005 (2022) [arXiv:2208.01604].

G. Bonelli, C. Iossa, D. P. Lichtig and A. Tanzini, Irregular Liouville correlators and connection formulae for Heun functions, Commun. Math. Phys. 397, 635 (2023) [arXiv:2201.04491].

Online resources

Heun Function in Wikipedia

Wolfram: Summary of Special Functions | All Mathematical Functions (long)