The HeunG function is the solution of the Heun General equation. Following the first reference (at the end), the equation and the conditions at the origin satisfied by HeunG are

FunctionAdvisor(definition, HeunG);

$\mathrm{HeunG}\left(a\,q\,\mathrm{\alpha }\,\mathrm{\beta }\,\mathrm{\gamma }\,\mathrm{\delta }\,z\right)=\mathrm{DESol}\left(\left\{\frac{{\ⅆ}^2}{\ⅆz^2}\mathrm{\_Y}\left(z\right)-\frac{\left(\left(\mathrm{\alpha }+\mathrm{\beta }+1\right)z^2+\left(\left(-\mathrm{\delta }-\mathrm{\gamma }\right)a-\mathrm{\alpha }+\mathrm{\delta }-\mathrm{\beta }-1\right)z+\mathrm{\gamma }a\right)\left(\frac{\ⅆ}{\ⅆz}\mathrm{\_Y}\left(z\right)\right)}{z\left(z-1\right)\left(-z+a\right)}-\frac{\left(\mathrm{\alpha }\mathrm{\beta }z-q\right)\mathrm{\_Y}\left(z\right)}{z\left(z-1\right)\left(-z+a\right)}\right\}\,\left\{\mathrm{\_Y}\left(z\right)\right\}\,\left\{\mathrm{\_Y}\left(0\right)=1\,\mathrm{D}\left(\mathrm{\_Y}\right)\left(0\right)=\frac{q}{\mathrm{\gamma }a}\right\}\right)$

Heun's equation is an extension of the 2F1 hypergeometric equation in that it is a second-order Fuchsian equation with four regular singular points. The 2F1 equation has three regular singularities. The HeunG function, thus, contains as particular cases all the functions of the hypergeometric 2F1 class.

Another important case of Heun's equation, for $\left\{\mathrm{\alpha }=-\frac{\mathrm{\nu }}{2}\,\mathrm{\beta }=\frac{1}{2}+\frac{\mathrm{\nu }}{2}\,\mathrm{\delta }=\frac{1}{2}\,\mathrm{\gamma }=\frac{1}{2}\,q=-\frac{ah}{4}\right\}$, is Lame's equation in algebraic form,

diff(y(z),z,z) + 1/2*(1/z + 1/(z-1) + 1/(z-a))*diff(y(z),z) + (a*h-nu*(nu+1)*z)/(4*z*(z-1)*(z-a))*y(z) = 0;

$\frac{{\ⅆ}^2}{\ⅆz^2}y\left(z\right)+\frac{\left(\frac{1}{z}+\frac{1}{z-1}+\frac{1}{z-a}\right)\left(\frac{\ⅆ}{\ⅆz}y\left(z\right)\right)}{2}+\frac{\left(ah-\mathrm{\nu }\left(1+\mathrm{\nu }\right)z\right)y\left(z\right)}{4z\left(z-1\right)\left(z-a\right)}=0$

where the parameter $\mathrm{\nu }$ is called the order of the equation and many special features arise when $\mathrm{\nu }$ is an integer. Lame's equation arises in the separation of variables in Laplace's equation.

The HeunG(a,q,$\mathrm{\alpha }$,$\mathrm{\beta }$,$\mathrm{\gamma }$,$\mathrm{\delta }$,z) function is a local (Frobenius) solution to Heun's equation, computed as a power series expansion around the origin, a regular singular point. The radius of convergence of this series is $\left|z_s\right|$ where $z_s$ is the location of the singularity closer to the origin, either a or 1. An analytic continuation of HeunG is obtained through identities, relating the values of the function in different regions, or by expanding the solution around the other singularities (a, 1 or $\mathrm{\infty }$). For example, the value of the function outside a circle of radius 2 is computable from the value of another HeunG function inside that circle using

HeunG(a,q,alpha,beta,gamma,delta,z) = (1-z)^(-alpha)*HeunG(a/(a-1),(-q+gamma*alpha*a)/(a-1),alpha,alpha-delta+1,gamma,alpha-beta+1,z/(z-1));

$\mathrm{HeunG}\left(a\,q\,\mathrm{\alpha }\,\mathrm{\beta }\,\mathrm{\gamma }\,\mathrm{\delta }\,z\right)={\left(1-z\right)}^{-\mathrm{\alpha }}\mathrm{HeunG}\left(\frac{a}{a-1}\,\frac{\mathrm{\gamma }\mathrm{\alpha }a-q}{a-1}\,\mathrm{\alpha }\,\mathrm{\alpha }-\mathrm{\delta }+1\,\mathrm{\gamma }\,\mathrm{\alpha }-\mathrm{\beta }+1\,\frac{z}{z-1}\right)$

For certain values of the parameters, it can happen that the expansion around the origin is also a Frobenius solution around the next adjacent singularity, so the function is analytic in some domain including both singularities. In the literature, the term Heun function is sometimes reserved for these cases, important in physical applications.

A more special situation happens when the parameters entering HeunG are such that the function is, simultaneously, a Frobenius solution around three adjacent singularities and hence analytic in a domain containing all of them. In such a case the solution will also be a Frobenius solution around the fourth singularity and HeunG will be a polynomial. A necessary (not sufficient) condition for this case is that $\mathrm{\alpha }=-n$, with $n$ a positive integer, and $q$ has one of a finite number of characteristic values, in which case the function is a polynomial of degree $n$.

$\mathrm{Matrix}\left(\left[\left[z\,1-z\,\frac{1}{z}\,\frac{1}{1-z}\right]\,\left[\frac{z}{z-1}\,\frac{z-1}{z}\,\frac{z}{a}\,\frac{a-z}{a}\right]\,\left[\frac{a}{z}\,\frac{a}{a-z}\,\frac{z}{z-a}\,\frac{z-a}{z}\right]\,\left[\frac{z-a}{1-a}\,\frac{z-1}{a-1}\,\frac{1-a}{z-a}\,\frac{a-1}{z-1}\right]\,\left[\frac{z-a}{z-1}\,\frac{z-1}{z-a}\,\frac{z-a}{a\left(z-1\right)}\,\frac{\left(a-1\right)z}{z-a}\right]\,\left[\frac{a\left(z-1\right)}{z-a}\,\frac{a\left(z-1\right)}{\left(a-1\right)z}\,\frac{z-a}{\left(1-a\right)z}\,\frac{\left(1-a\right)z}{z-a}\right]\right]\right)$

$\left[\begin{array}{cccc}z& 1-z& \frac{1}{z}& \frac{1}{1-z}\\ \frac{z}{z-1}& \frac{z-1}{z}& \frac{z}{a}& \frac{a-z}{a}\\ \frac{a}{z}& \frac{a}{a-z}& \frac{z}{z-a}& \frac{z-a}{z}\\ \frac{z-a}{1-a}& \frac{z-1}{a-1}& \frac{1-a}{z-a}& \frac{a-1}{z-1}\\ \frac{z-a}{z-1}& \frac{z-1}{z-a}& \frac{z-a}{a\left(z-1\right)}& \frac{\left(a-1\right)z}{z-a}\\ \frac{a\left(z-1\right)}{z-a}& \frac{a\left(z-1\right)}{\left(a-1\right)z}& \frac{z-a}{\left(1-a\right)z}& \frac{\left(1-a\right)z}{z-a}\end{array}\right]$

$\mathrm{FunctionAdvisor}\left(\mathrm{identities}\,\mathrm{HeunG}\right)$

$\left[\mathrm{HeunG}\left(a\,q\,\mathrm{\alpha }\,\mathrm{\beta }\,\mathrm{\gamma }\,\mathrm{\delta }\,z\right)={\left(1-z\right)}^{1-\mathrm{\delta }}\mathrm{HeunG}\left(a\,q-\left(\mathrm{\delta }-1\right)\mathrm{\gamma }a\,\mathrm{\beta }-\mathrm{\delta }+1\,\mathrm{\alpha }-\mathrm{\delta }+1\,\mathrm{\gamma }\,2-\mathrm{\delta }\,z\right)\,\left[\mathrm{HeunG}\left(a\,q\,\mathrm{\alpha }\,\mathrm{\beta }\,\mathrm{\gamma }\,\mathrm{\delta }\,z\right)={\left(1-\frac{z}{a}\right)}^{-\mathrm{\alpha }-\mathrm{\beta }+\mathrm{\gamma }+\mathrm{\delta }}\mathrm{HeunG}\left(a\,q-\mathrm{\gamma }\left(\mathrm{\alpha }+\mathrm{\beta }-\mathrm{\gamma }-\mathrm{\delta }\right)\,-\mathrm{\beta }+\mathrm{\gamma }+\mathrm{\delta }\,-\mathrm{\alpha }+\mathrm{\gamma }+\mathrm{\delta }\,\mathrm{\gamma }\,\mathrm{\delta }\,z\right)\,a\ne 0\right]\,\left[\mathrm{HeunG}\left(a\,q\,\mathrm{\alpha }\,\mathrm{\beta }\,\mathrm{\gamma }\,\mathrm{\delta }\,z\right)={\left(1-z\right)}^{1-\mathrm{\delta }}{\left(1-\frac{z}{a}\right)}^{-\mathrm{\alpha }-\mathrm{\beta }+\mathrm{\gamma }+\mathrm{\delta }}\mathrm{HeunG}\left(a\,q-\mathrm{\gamma }\left(\left(\mathrm{\delta }-1\right)a+\mathrm{\alpha }+\mathrm{\beta }-\mathrm{\gamma }-\mathrm{\delta }\right)\,-\mathrm{\beta }+\mathrm{\gamma }+1\,-\mathrm{\alpha }+\mathrm{\gamma }+1\,\mathrm{\gamma }\,2-\mathrm{\delta }\,z\right)\,a\ne 0\right]\,\left[\mathrm{HeunG}\left(a\,q\,\mathrm{\alpha }\,\mathrm{\beta }\,\mathrm{\gamma }\,\mathrm{\delta }\,z\right)=\mathrm{HeunG}\left(\frac{1}{a}\,\frac{q}{a}\,\mathrm{\alpha }\,\mathrm{\beta }\,\mathrm{\gamma }\,\mathrm{\alpha }+\mathrm{\beta }-\mathrm{\gamma }-\mathrm{\delta }+1\,\frac{z}{a}\right)\,a\ne 0\right]\,\left[\mathrm{HeunG}\left(a\,q\,\mathrm{\alpha }\,\mathrm{\beta }\,\mathrm{\gamma }\,\mathrm{\delta }\,z\right)={\left(1-z\right)}^{-\mathrm{\alpha }}\mathrm{HeunG}\left(\frac{a}{a-1}\,\frac{\mathrm{\alpha }\mathrm{\gamma }a-q}{a-1}\,\mathrm{\alpha }\,\mathrm{\alpha }-\mathrm{\delta }+1\,\mathrm{\gamma }\,\mathrm{\alpha }-\mathrm{\beta }+1\,\frac{z}{z-1}\right)\,a\ne 1\wedge z\ne 1\right]\,\left[\mathrm{HeunG}\left(a\,q\,\mathrm{\alpha }\,\mathrm{\beta }\,\mathrm{\gamma }\,\mathrm{\delta }\,z\right)={\left(1-\frac{z}{a}\right)}^{-\mathrm{\alpha }}\mathrm{HeunG}\left(1-a\,\mathrm{\gamma }\mathrm{\alpha }-q\,\mathrm{\alpha }\,-\mathrm{\beta }+\mathrm{\gamma }+\mathrm{\delta }\,\mathrm{\gamma }\,\mathrm{\delta }\,\frac{\left(1-a\right)z}{z-a}\right)\,a\ne 0\wedge z\ne a\right]\,\left[\mathrm{HeunG}\left(a\,q\,\mathrm{\alpha }\,\mathrm{\beta }\,\mathrm{\gamma }\,\mathrm{\delta }\,z\right)={\left(1-\frac{z}{a}\right)}^{-\mathrm{\alpha }}\mathrm{HeunG}\left(\frac{1}{1-a}\,\frac{-\mathrm{\gamma }\mathrm{\alpha }+q}{a-1}\,\mathrm{\alpha }\,-\mathrm{\beta }+\mathrm{\gamma }+\mathrm{\delta }\,\mathrm{\gamma }\,\mathrm{\alpha }-\mathrm{\beta }+1\,\frac{z}{z-a}\right)\,a\ne 0\wedge a\ne 1\wedge z\ne a\right]\right]$

$\mathrm{FunctionAdvisor}\left(\mathrm{specialize}\,\mathrm{HeunG}\,\mathrm{hypergeom}\right)$

$\left[\mathrm{HeunG}\left(a\&comma;q\&comma;\mathrm{\alpha }\&comma;\mathrm{\beta }\&comma;\mathrm{\gamma }\&comma;\mathrm{\delta }\&comma;z\right)=\mathrm{hypergeom}\left(\left[\mathrm{\alpha }\&comma;\mathrm{\beta }\right]\&comma;\left[\mathrm{\alpha }-\mathrm{\delta }+1+\mathrm{\beta }\right]\&comma;z\right)\&comma;a=0\wedge q=0\right],\left[\mathrm{HeunG}\left(a\&comma;q\&comma;\mathrm{\alpha }\&comma;\mathrm{\beta }\&comma;\mathrm{\gamma }\&comma;\mathrm{\delta }\&comma;z\right)=\mathrm{hypergeom}\left(\left[\mathrm{\alpha }\&comma;\mathrm{\beta }\right]\&comma;\left[\mathrm{\gamma }\right]\&comma;z\right)\&comma;\left(a=1\wedge q=\mathrm{\alpha }\mathrm{\beta }\right)\vee \left(q=a\mathrm{\alpha }\mathrm{\beta }\wedge \mathrm{\delta }=\mathrm{\alpha }+\mathrm{\beta }-\mathrm{\gamma }+1\right)\right],\left[\mathrm{HeunG}\left(a\&comma;q\&comma;\mathrm{\alpha }\&comma;\mathrm{\beta }\&comma;\mathrm{\gamma }\&comma;\mathrm{\delta }\&comma;z\right)=\frac{\mathrm{hypergeom}\left(\left[\frac{\sqrt{{\mathrm{\gamma }}^2+2\left(-\mathrm{\alpha }-\mathrm{\beta }\right)\mathrm{\gamma }+{\left(\mathrm{\alpha }-\mathrm{\beta }\right)}^2+4q}}{2}+\frac{\mathrm{\gamma }}{2}+\frac{\mathrm{\alpha }}{2}-\frac{\mathrm{\beta }}{2}\&comma;\frac{\sqrt{{\mathrm{\gamma }}^2+2\left(-\mathrm{\alpha }-\mathrm{\beta }\right)\mathrm{\gamma }+{\left(\mathrm{\alpha }-\mathrm{\beta }\right)}^2+4q}}{2}+\frac{\mathrm{\gamma }}{2}-\frac{\mathrm{\alpha }}{2}+\frac{\mathrm{\beta }}{2}\right]\&comma;\left[\mathrm{\gamma }\right]\&comma;z\right)}{{\left(z-1\right)}^{-\frac{\sqrt{{\mathrm{\gamma }}^2+2\left(-\mathrm{\alpha }-\mathrm{\beta }\right)\mathrm{\gamma }+{\left(\mathrm{\alpha }-\mathrm{\beta }\right)}^2+4q}}{2}-\frac{\mathrm{\gamma }}{2}+\frac{\mathrm{\alpha }}{2}+\frac{\mathrm{\beta }}{2}}{\&ExponentialE;}^{\frac{\mathrm{I}}{2}\mathrm{\pi }\left(\sqrt{{\mathrm{\gamma }}^2+2\left(-\mathrm{\alpha }-\mathrm{\beta }\right)\mathrm{\gamma }+{\left(\mathrm{\alpha }-\mathrm{\beta }\right)}^2+4q}+\mathrm{\gamma }-\mathrm{\alpha }-\mathrm{\beta }\right)}}\&comma;a=1\wedge 0\le \mathrm{\Im }\left(z\right)\right],\left[\mathrm{HeunG}\left(a\&comma;q\&comma;\mathrm{\alpha }\&comma;\mathrm{\beta }\&comma;\mathrm{\gamma }\&comma;\mathrm{\delta }\&comma;z\right)=\mathrm{hypergeom}\left(\left[\frac{\mathrm{\beta }}{2}\&comma;\frac{\mathrm{\alpha }}{2}\right]\&comma;\left[\mathrm{\gamma }\right]\&comma;-z\left(z-2\right)\right)\&comma;a=2\wedge q=\mathrm{\alpha }\mathrm{\beta }\wedge \mathrm{\delta }=\mathrm{\alpha }+\mathrm{\beta }-2\mathrm{\gamma }+1\wedge \mathrm{-1}<\mathrm{\Re }\left(z\right)\wedge \mathrm{\Re }\left(z\right)<1\right],\left[\mathrm{HeunG}\left(a\&comma;q\&comma;\mathrm{\alpha }\&comma;\mathrm{\beta }\&comma;\mathrm{\gamma }\&comma;\mathrm{\delta }\&comma;z\right)=\mathrm{hypergeom}\left(\left[\frac{\mathrm{\beta }}{3}\&comma;\frac{\mathrm{\alpha }}{3}\right]\&comma;\left[\frac{1}{2}\right]\&comma;\frac{z{\left(z-3\right)}^2}{4}\right)\&comma;a=4\wedge q=\mathrm{\alpha }\mathrm{\beta }\wedge \mathrm{\gamma }=\frac{1}{2}\wedge \mathrm{\delta }=\frac{2\mathrm{\alpha }}{3}+\frac{2\mathrm{\beta }}{3}\wedge \mathrm{-1}<\mathrm{\Re }\left(z\right)\wedge \mathrm{\Re }\left(z\right)<1\wedge -\mathrm{\pi }<\mathrm{\Im }\left(z\right)\wedge \mathrm{\Im }\left(z\right)<\mathrm{\pi }\right]$

$\mathrm{HG}≔\mathrm{HeunG}\left(a\&comma;q\&comma;-1\&comma;\mathrm{\beta }\&comma;g\&comma;\mathrm{\delta }\&comma;z\right)$

$\mathrm{HG}≔\mathrm{HeunG}\left(a\&comma;q\&comma;\mathrm{-1}\&comma;\mathrm{\beta }\&comma;g\&comma;\mathrm{\delta }\&comma;z\right)$

$Q≔\mathrm{simplify}\left(\mathrm{series}\left(\mathrm{HG}\&comma;z\&comma;3\right)\&comma;\mathrm{size}\right)$

$Q≔1+\frac{q}{ga}z+\frac{1}{2}\frac{q^2+\left(\left(g+\mathrm{\delta }\right)a+\mathrm{\beta }-\mathrm{\delta }\right)q+\mathrm{\beta }ga}{g{a}^2\left(g+1\right)}{z}^2+O\left(z^3\right)$

$\mathrm{c2}≔\mathrm{coeff}\left(Q\&comma;z\&comma;2\right)$

$\mathrm{c2}≔\frac{q^2+\left(\left(g+\mathrm{\delta }\right)a+\mathrm{\beta }-\mathrm{\delta }\right)q+\mathrm{\beta }ga}{2g{a}^2\left(g+1\right)}$

$\mathrm{\_EnvExplicit}≔\mathrm{false}$

$\mathrm{\_EnvExplicit}≔\mathrm{false}$

$q=\mathrm{solve}\left(\mathrm{c2}\&comma;q\right)$

$q=\mathrm{RootOf}\left({\mathrm{\_Z}}^2+\left(a\mathrm{\delta }+ga+\mathrm{\beta }-\mathrm{\delta }\right)\mathrm{\_Z}+\mathrm{\beta }ga\right)$

$\mathrm{HG\_polynomial}≔\mathrm{subs}\left(\&comma;\mathrm{HG}\right)$

$\mathrm{HG\_polynomial}≔\mathrm{HeunG}\left(a\&comma;\mathrm{RootOf}\left({\mathrm{\_Z}}^2+\left(a\mathrm{\delta }+ga+\mathrm{\beta }-\mathrm{\delta }\right)\mathrm{\_Z}+\mathrm{\beta }ga\right)\&comma;\mathrm{-1}\&comma;\mathrm{\beta }\&comma;g\&comma;\mathrm{\delta }\&comma;z\right)$

$\mathrm{eval}\left(\&comma;\mathrm{HeunG}=\mathrm{HeunG}:-\mathrm{SpecialValues}:-\mathrm{Polynomial}\right)$

$1+\frac{\mathrm{RootOf}\left({\mathrm{\_Z}}^2+\left(a\mathrm{\delta }+ga+\mathrm{\beta }-\mathrm{\delta }\right)\mathrm{\_Z}+\mathrm{\beta }ga\right)z}{ga}$

Ronveaux, A. ed. Heun's Differential Equations. Oxford University Press, 1995.

Slavyanov, S.Y., and Lay, W. Special Functions, A Unified Theory Based on Singularities. Oxford Mathematical Monographs, 2000.