The General Heun equation, with four regular singular points, is

with(DEtools,singularities);

$\left[\mathrm{singularities}\right]$

PDEtools[declare](y(z), prime=z, quiet);

GHE := diff(y(z),z,z) + (gamma/z+delta/(z-1)+epsilon/(z-a))*diff(y(z),z) + (alpha*beta*z-q)/z/(z-1)/(z-a)*y(z) = 0;

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

singularities( GHE );

$\mathrm{regular}=\left\{0\,1\,a\,\mathrm{\infty }\right\},\mathrm{irregular}=\varnothing$

The solution to this equation is implemented in Maple as the HeunG function. The sum of the exponents of the singularities of Heun's equation is equal to two and the parameter $\mathrm{\epsilon }$ is expressed in terms of the other ones by

epsilon = alpha+beta+1 -gamma-delta;

$\mathrm{\epsilon }=\mathrm{\alpha }+\mathrm{\beta }+1-\mathrm{\gamma }-\mathrm{\delta }$

The choice of standard form for the GHE is uniform in the literature except in one particular: the numerator of the coefficient of $y$ is sometimes presented as $\mathrm{\alpha }\mathrm{\beta }\left(z-q\right)$. The Maple choice of standard form, and so the definition of HeunG implemented, follows Chapter 1 in [2] (see references at the end) and has the small but significant advantage that you can take $\mathrm{\alpha }=0$ (or $\mathrm{\beta }=0$) without eliminating the term in $y$ completely.

The other four Heun equations are confluent cases, obtained from the general Heun equation above through confluence processes. These are: the Heun Confluent equation

CHE := diff(y(z),z,z) + (gamma/z+delta/(z-1)-epsilon)*diff(y(z),z) +((q-alpha*beta)/(z-1)-q/z)*y(z) = 0;

$\mathrm{CHE}≔\mathrm{y\text{'}\text{'}}+\left(\frac{\mathrm{\gamma }}{z}+\frac{\mathrm{\delta }}{z-1}-\mathrm{\epsilon }\right)\mathrm{y\text{'}}+\left(\frac{-\mathrm{\alpha }\mathrm{\beta }+q}{z-1}-\frac{q}{z}\right)y=0$

singularities( CHE );

$\mathrm{regular}=\left\{0\,1\right\},\mathrm{irregular}=\left\{\mathrm{\infty }\right\}$

having for solution the HeunC function; the Biconfluent equation

BHE := diff(y(z),z,z) + (-2*z-beta+(1+alpha)/z)*diff(y(z),z) + (gamma-alpha-2 - 1/2*((1+alpha)*beta+delta)/z)*y(z) = 0;

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

singularities( BHE );

$\mathrm{regular}=\left\{0\right\},\mathrm{irregular}=\left\{\mathrm{\infty }\right\}$

having for solution the HeunB function; the Doubleconfluent equation

DHE := diff(y(z),z,z)-(alpha+2*z+z^2*alpha-2*z^3)/(z+1)^2/(z-1)^2*diff(y(z),z)+(delta+(2*alpha+gamma)*z+beta*z^2)/(z-1)^3/(z+1)^3*y(z);

$\mathrm{DHE}≔\mathrm{y\text{'}\text{'}}-\frac{\left(\mathrm{\alpha }{z}^2-2z^3+\mathrm{\alpha }+2z\right)\mathrm{y\text{'}}}{{\left(z+1\right)}^2{\left(z-1\right)}^2}+\frac{\left(\mathrm{\delta }+\left(2\mathrm{\alpha }+\mathrm{\gamma }\right)z+\mathrm{\beta }{z}^2\right)y}{{\left(z-1\right)}^3{\left(z+1\right)}^3}$

singularities( DHE );

$\mathrm{regular}=\varnothing ,\mathrm{irregular}=\left\{\mathrm{-1}\,1\right\}$

having for solution the HeunD function, and the Triconfluent equation

THE := diff(y(z),z,z) + (-gamma-3*z^2)*diff(y(z),z) + (alpha+z*beta-3*z)*y(z) = 0;

$\mathrm{THE}≔\mathrm{y\text{'}\text{'}}+\left(-3z^2-\mathrm{\gamma }\right)\mathrm{y\text{'}}+\left(\mathrm{\beta }z+\mathrm{\alpha }-3z\right)y=0$

singularities( THE );

$\mathrm{regular}=\varnothing ,\mathrm{irregular}=\left\{\mathrm{\infty }\right\}$

having for solution the HeunT function.

The standard form of the four confluent equations is not uniform in the literature. The Maple choice of standard form follows [1], the classic reference for these equations, except for one particular: for the Doubleconfluent equation (DHE above) (and so the definition of HeunD) Maple uses the so-called Jaffe form, see [3], so that the two irregular singular points are symmetrically located at $z=\mathrm{-1}$ and $z=1$.

The Heun functions, HeunG, HeunC, HeunB, HeunD and HeunT, are defined as the solutions to the corresponding General, Confluent, Biconfluent, Doubleconfluent and Triconfluent Heun equations. These solutions are constructed as power series solutions around the origin, for certain initial conditions. The five equations and the initial conditions used to define the corresponding five Heun functions can be seen through the FunctionAdvisor

FunctionAdvisor(definition,HeunG);

FunctionAdvisor(definition,HeunC);

$\mathrm{HeunC}\left(\mathrm{\alpha }\,\mathrm{\beta }\,\mathrm{\gamma }\,\mathrm{\delta }\,\mathrm{\eta }\,z\right)=\mathrm{DESol}\left(\left\{{\mathrm{\_Y}}_{z,z}-\frac{\left(-z^2\mathrm{\alpha }+\left(-\mathrm{\beta }+\mathrm{\alpha }-\mathrm{\gamma }-2\right)z+\mathrm{\beta }+1\right){\mathrm{\_Y}}_z}{z\left(z-1\right)}-\frac{\left(\left(\left(-\mathrm{\beta }-\mathrm{\gamma }-2\right)\mathrm{\alpha }-2\mathrm{\delta }\right)z+\left(\mathrm{\beta }+1\right)\mathrm{\alpha }+\left(-\mathrm{\gamma }-1\right)\mathrm{\beta }-2\mathrm{\eta }-\mathrm{\gamma }\right)\mathrm{\_Y}\left(z\right)}{2z\left(z-1\right)}\right\}\,\left\{\mathrm{\_Y}\left(z\right)\right\}\,\left\{\mathrm{\_Y}\left(0\right)=1\,\mathrm{D}\left(\mathrm{\_Y}\right)\left(0\right)=\frac{\left(-\mathrm{\alpha }+1+\mathrm{\gamma }\right)\mathrm{\beta }+\mathrm{\gamma }-\mathrm{\alpha }+2\mathrm{\eta }}{2\left(\mathrm{\beta }+1\right)}\right\}\right)$

FunctionAdvisor(definition,HeunB);

FunctionAdvisor(definition,HeunD);

FunctionAdvisor(definition,HeunT);

The power series solutions at the base of the functions' definitions have restricted radius of convergence in the HeunG, HeunC and HeunD cases, where the numerical evaluation is done using analytic extensions, exploring closed form identities satisfied by these functions, as well as series expansions around different singularities. For arbitrary values of the parameters, however, closed form formulas for the connection constants relating series expansions around different singularities, are not known.

As an example of the use of identities, for HeunD,

FunctionAdvisor(identities,HeunD);

$\left[\mathrm{HeunD}\left(\mathrm{\alpha }\,\mathrm{\beta }\,\mathrm{\gamma }\,\mathrm{\delta }\,z\right)=\mathrm{HeunD}\left(-\mathrm{\alpha }\,-\mathrm{\delta }\,-\mathrm{\gamma }\,-\mathrm{\beta }\,\frac{1}{z}\right)\right]$

Hence, the evaluation of the function in the complex plane outside the unit circle is performed by mapping the problem into that of evaluating the function inside that circle. A similar situation is described in the help page for HeunG.

Remark: The coefficients entering the series expansions represented by the Heun functions satisfy three term recurrence relations. A solution to these recursion equations is not known in the general case, so a closed form for the series's coefficients is not available and the computation of - say - the $n$th coefficient requires the explicit computation of all the previous ones.