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

FunctionAdvisor(definition, HeunT);

$\mathrm{HeunT}\left(\mathrm{\alpha }\,\mathrm{\beta }\,\mathrm{\gamma }\,z\right)=\mathrm{DESol}\left(\left\{\frac{{\ⅆ}^2}{\ⅆz^2}\mathrm{\_Y}\left(z\right)-\left(3z^2+\mathrm{\gamma }\right)\left(\frac{\ⅆ}{\ⅆz}\mathrm{\_Y}\left(z\right)\right)-\left(\left(-\mathrm{\beta }+3\right)z-\mathrm{\alpha }\right)\mathrm{\_Y}\left(z\right)\right\}\,\left\{\mathrm{\_Y}\left(z\right)\right\}\,\left\{\mathrm{\_Y}\left(0\right)=1\,\mathrm{D}\left(\mathrm{\_Y}\right)\left(0\right)=0\right\}\right)$

The HeunT($\mathrm{\alpha }$,$\mathrm{\beta }$,$\mathrm{\gamma }$,z) function is a local solution to Heun's Triconfluent equation, computed as a standard power series expansion around the origin, a regular point. Because the single singularity is located at $\mathrm{\infty }$, this series converges in the whole complex plane.

The Triconfluent Heun Equation (THE) above is obtained from the Doubleconfluent Heun Equation (DHE) through a confluence process, that is, a process where two singularities coalesce, performed by redefining parameters and taking limits. In this case the two irregular singularities of the DHE are coalesced into one irregular singularity at $\mathrm{\infty }$. The resulting Heun Triconfluent equation, thus, has the structure of singularities f of the 0F1 hypergeometric equation and so can be related to the Airy functions.

A special case happens when in HeunT($\mathrm{\alpha }$,$\mathrm{\beta }$,$\mathrm{\gamma }$,z), the second parameter satisfies $\mathrm{\beta }=3\left(n+1\right)$, where $n$ is a positive integer. In this case the $n$th+1, $n$th+2 and $n$th+3 coefficients form a polynomial system for the remaining parameters $\mathrm{\alpha }$ and $\mathrm{\gamma }$; when this system is identically satisfied all the subsequent coefficients cancel too and the series truncates, resulting in a polynomial form of degree $n$ for HeunT. Remark: for $n=0$ this situation leads to a constant, for $n=1$ HeunT will also be a constant since its series expansion satisfies $\mathrm{HeunT}\text{'}$ at 0 = 0 and for $n=2$ the polynomial system for $\mathrm{\alpha }$ and $\mathrm{\gamma }$ is inconsistent. So the non-trivial polynomial forms of HeunT are of degree $3\le n$.

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

$\mathrm{THE}≔\frac{{\ⅆ}^2}{\ⅆz^2}y\left(z\right)=\left(3z^2+\mathrm{\gamma }\right)\left(\frac{\ⅆ}{\ⅆz}y\left(z\right)\right)+\left(\left(3-\mathrm{\beta }\right)z-\mathrm{\alpha }\right)y\left(z\right)$

$z=\mathrm{\kappa }t,y\left(z\right)=\exp \left(\frac{\frac{1}{2}t\left(\mathrm{\kappa }-1\right)\left({\mathrm{\kappa }}^2+\mathrm{\kappa }+1\right)\left(\mathrm{\gamma }+t^2\right)}{{\mathrm{\kappa }}^3}\right)u\left(t\right)$

$z=\mathrm{\kappa }t,y\left(z\right)={\ⅇ}^{\frac{t\left(\mathrm{\kappa }-1\right)\left({\mathrm{\kappa }}^2+\mathrm{\kappa }+1\right)\left(t^2+\mathrm{\gamma }\right)}{2{\mathrm{\kappa }}^3}}u\left(t\right)$

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

$\left[\left[\mathrm{HeunT}\left(\mathrm{\alpha }\,\mathrm{\beta }\,\mathrm{\gamma }\,z\right)=\mathrm{HeunT}\left(j\mathrm{\alpha }\,\mathrm{\beta }\,j^2\mathrm{\gamma }\,jz\right)\,j^3=1\right]\,\left[\mathrm{HeunT}\left(\mathrm{\alpha }\,\mathrm{\beta }\,\mathrm{\gamma }\,z\right)=\mathrm{HeunT}\left(\mathrm{\alpha }\,-\mathrm{\beta }\,\mathrm{\gamma }\,-z\right){\ⅇ}^{z^3}\,\mathrm{\gamma }=0\right]\right]$

$\mathrm{HeunT}\left(\mathrm{\alpha }\,3n+3\,\mathrm{\gamma }\,z\right)$

$\mathrm{HeunT}\left(\mathrm{\alpha }\,3n+3\,\mathrm{\gamma }\,z\right)$

$\mathrm{HT}≔\mathrm{subs}\left(n=3\,\right)$

$\mathrm{HT}≔\mathrm{HeunT}\left(\mathrm{\alpha }\,12\,\mathrm{\gamma }\,z\right)$

$Q≔\mathrm{simplify}\left(\mathrm{series}\left(\mathrm{HT}\,z\,7\right)\,\mathrm{size}\right)$

$Q≔1-\frac{1}{2}\mathrm{\alpha }{z}^2+\left(-\frac{\mathrm{\gamma }\mathrm{\alpha }}{6}-\frac{3}{2}\right)z^3+\left(\frac{1}{24}{\mathrm{\alpha }}^2-\frac{1}{24}{\mathrm{\gamma }}^2\mathrm{\alpha }-\frac{3}{8}\mathrm{\gamma }\right)z^4-\frac{1}{120}\left(\mathrm{\gamma }\mathrm{\alpha }+9\right)\left({\mathrm{\gamma }}^2-2\mathrm{\alpha }\right)z^5+\left(-\frac{{\mathrm{\alpha }}^3}{720}+\frac{{\mathrm{\gamma }}^2{\mathrm{\alpha }}^2}{240}+\frac{\left(-{\mathrm{\gamma }}^4+27\mathrm{\gamma }\right)\mathrm{\alpha }}{720}-\frac{{\mathrm{\gamma }}^3}{80}\right)z^6+O\left(z^7\right)$

$\mathrm{c4},\mathrm{c5},\mathrm{c6}≔\mathrm{coeff}\left(Q\,z\,4\right),\mathrm{coeff}\left(Q\,z\,5\right),\mathrm{coeff}\left(Q\,z\,6\right)$

$\mathrm{c4},\mathrm{c5},\mathrm{c6}≔\frac{1}{24}{\mathrm{\alpha }}^2-\frac{1}{24}{\mathrm{\gamma }}^2\mathrm{\alpha }-\frac{3}{8}\mathrm{\gamma },-\frac{\left(\mathrm{\gamma }\mathrm{\alpha }+9\right)\left({\mathrm{\gamma }}^2-2\mathrm{\alpha }\right)}{120},-\frac{{\mathrm{\alpha }}^3}{720}+\frac{{\mathrm{\gamma }}^2{\mathrm{\alpha }}^2}{240}+\frac{\left(-{\mathrm{\gamma }}^4+27\mathrm{\gamma }\right)\mathrm{\alpha }}{720}-\frac{{\mathrm{\gamma }}^3}{80}$

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

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

$\mathrm{subs}\left(\mathrm{ga}=\mathrm{\gamma }\,\left[\mathrm{solve}\left(\mathrm{subs}\left(\mathrm{\gamma }=\mathrm{ga}\,\left\{\mathrm{c4}\,\mathrm{c5}\,\mathrm{c6}\right\}\right)\,\left\{\mathrm{\alpha }\,\mathrm{ga}\right\}\right)\right]\right)$

$\left[\left\{\mathrm{\alpha }=0\,\mathrm{\gamma }=0\right\}\,\left\{\mathrm{\alpha }=\frac{{\mathrm{RootOf}\left({\mathrm{\_Z}}^3+36\right)}^2}{2}\,\mathrm{\gamma }=\mathrm{RootOf}\left({\mathrm{\_Z}}^3+36\right)\right\}\right]$

$\mathrm{HT\_polynomial}≔\mathrm{subs}\left(\left[1\right]\,\mathrm{HT}\right)$

$\mathrm{HT\_polynomial}≔\mathrm{HeunT}\left(0\,12\,0\,z\right)$

$\mathrm{eval}\left(\,\mathrm{HeunT}=\mathrm{HeunT}:-\mathrm{SpecialValues}:-\mathrm{Polynomial}\right)$

$1-\frac{3z^3}{2}$

Decarreau, A.; Dumont-Lepage, M.C.; Maroni, P.; Robert, A.; and Ronveaux, A. "Formes Canoniques de Equations confluentes de l'equation de Heun". Annales de la Societe Scientifique de Bruxelles. Vol. 92 I-II, (1978): 53-78.

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.