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

FunctionAdvisor(definition, HeunB);

$\mathrm{HeunB}\left(\mathrm{\alpha }\,\mathrm{\beta }\,\mathrm{\gamma }\,\mathrm{\delta }\,z\right)=\mathrm{DESol}\left(\left\{\frac{{\ⅆ}^2}{\ⅆz^2}\mathrm{\_Y}\left(z\right)-\frac{\left(\mathrm{\beta }z+2z^2-\mathrm{\alpha }-1\right)\left(\frac{\ⅆ}{\ⅆz}\mathrm{\_Y}\left(z\right)\right)}{z}-\frac{\left(\left(2\mathrm{\alpha }-2\mathrm{\gamma }+4\right)z+\mathrm{\beta }\mathrm{\alpha }+\mathrm{\beta }+\mathrm{\delta }\right)\mathrm{\_Y}\left(z\right)}{2z}\right\}\,\left\{\mathrm{\_Y}\left(z\right)\right\}\,\left\{\mathrm{\_Y}\left(0\right)=1\,\mathrm{D}\left(\mathrm{\_Y}\right)\left(0\right)=\frac{\mathrm{\beta }\mathrm{\alpha }+\mathrm{\beta }+\mathrm{\delta }}{2\mathrm{\alpha }+2}\right\}\right)$

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

The Biconfluent Heun Equation (BHE) above is obtained from the Confluent Heun Equation (CHE) through a confluence process, that is, a process where two singularities coalesce, performed by redefining parameters and taking limits. In this case one regular singularity of the CHE is coalesced with its irregular singularity at $\mathrm{\infty }$. The resulting Heun Biconfluent equation, thus, has one regular singularity at the origin, one irregular one at $\mathrm{\infty }$, and includes as a particular case the 1F1 hypergeometric confluent equation

DEtools[hyperode]( hypergeom([a],[c],z), y(z) ) = 0;

$ay\left(z\right)+\left(-c+z\right)\left(\frac{\ⅆ}{\ⅆz}y\left(z\right)\right)-z\left(\frac{{\ⅆ}^2}{\ⅆz^2}y\left(z\right)\right)=0$

So besides the standard hypergeometric solution of this equation, a solution expressed in terms of HeunB functions can also be constructed, and in this way HeunB contains as particular cases all the hypergeometric functions of the 1F1 class. Some of these specializations are listed at the end of the Examples section.

A special case happens when in HeunB($\mathrm{\alpha }$, $\mathrm{\beta }$, $\mathrm{\gamma }$, $\mathrm{\delta }$, z) the third parameter satisfies $\mathrm{\gamma }=2\left(n+1\right)+\mathrm{\alpha }$, where $n$ is a positive integer. In this case the $n+1$th coefficient in the series expansion is a polynomial of degree $n$ in $\mathrm{\delta }$. When $\mathrm{\delta }$ is a root of this polynomial, the $n+1$th and subsequent coefficients cancel and the series truncates, resulting in a polynomial form of degree $n$ for HeunB.

$\mathrm{BHE}≔\mathrm{diff}\left(y\left(z\right)\,z\,z\right)=\frac{2z^2-1-\mathrm{\alpha }+\mathrm{\beta }z}{z}\mathrm{diff}\left(y\left(z\right)\,z\right)+\frac{\frac{1}{2}\left(\left(-2\mathrm{\gamma }+2\mathrm{\alpha }+4\right)z+\mathrm{\delta }+\mathrm{\beta }+\mathrm{\beta }\mathrm{\alpha }\right)}{z}y\left(z\right)$

$\mathrm{BHE}≔\frac{{\ⅆ}^2}{\ⅆz^2}y\left(z\right)=\frac{\left(\mathrm{\beta }z+2z^2-\mathrm{\alpha }-1\right)\left(\frac{\ⅆ}{\ⅆz}y\left(z\right)\right)}{z}+\frac{\left(\left(-2\mathrm{\gamma }+2\mathrm{\alpha }+4\right)z+\mathrm{\delta }+\mathrm{\beta }+\mathrm{\beta }\mathrm{\alpha }\right)y\left(z\right)}{2z}$

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

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

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

$\left[\left[\mathrm{HeunB}\left(\mathrm{\alpha }\,\mathrm{\beta }\,\mathrm{\gamma }\,\mathrm{\delta }\,z\right)=z^{-\mathrm{\alpha }}\mathrm{HeunB}\left(-\mathrm{\alpha }\,\mathrm{\beta }\,\mathrm{\gamma }\,\mathrm{\delta }\,z\right)\,\mathrm{\alpha }::{\mathbb{Z}}^{-}\wedge z\ne 0\right]\,\left[\mathrm{HeunB}\left(\mathrm{\alpha }\,\mathrm{\beta }\,\mathrm{\gamma }\,\mathrm{\delta }\,z\right)={\mathrm{I}}^{-\mathrm{\alpha }}{\ⅇ}^{\mathrm{\beta }z+z^2}\mathrm{HeunB}\left(\mathrm{\alpha }\,\mathrm{-I}\mathrm{\beta }\,-\mathrm{\gamma }\,\mathrm{I}\mathrm{\delta }\,\mathrm{-I}z\right)\,\mathrm{\alpha }::{\mathbb{Z}}^{-}\wedge z\ne 0\right]\,\left[\mathrm{HeunB}\left(\mathrm{\alpha }\,\mathrm{\beta }\,\mathrm{\gamma }\,\mathrm{\delta }\,z\right)={\left(\mathrm{-1}\right)}^{-\mathrm{\alpha }}\mathrm{HeunB}\left(\mathrm{\alpha }\,-\mathrm{\beta }\,\mathrm{\gamma }\,-\mathrm{\delta }\,-z\right)\,\mathrm{\alpha }::{\mathbb{Z}}^{-}\wedge z\ne 0\right]\,\left[\mathrm{HeunB}\left(\mathrm{\alpha }\,\mathrm{\beta }\,\mathrm{\gamma }\,\mathrm{\delta }\,z\right)={\ⅇ}^{\mathrm{\beta }z+z^2}\mathrm{HeunB}\left(\mathrm{\alpha }\,\mathrm{-I}\mathrm{\beta }\,-\mathrm{\gamma }\,\mathrm{I}\mathrm{\delta }\,\mathrm{-I}z\right)\,\mathrm{\alpha }::\left(\neg {\mathbb{Z}}^{-}\right)\right]\,\left[\mathrm{HeunB}\left(\mathrm{\alpha }\,\mathrm{\beta }\,\mathrm{\gamma }\,\mathrm{\delta }\,z\right)=\mathrm{HeunB}\left(\mathrm{\alpha }\,-\mathrm{\beta }\,\mathrm{\gamma }\,-\mathrm{\delta }\,-z\right)\,\mathrm{\alpha }::\left(\neg {\mathbb{Z}}^{-}\right)\right]\right]$

$\left[\mathrm{FunctionAdvisor}\left(\mathrm{specialize}\,\mathrm{HeunB}\,\mathrm{hypergeom}\right)\right]$

$\left[\left[\mathrm{HeunB}\left(\mathrm{\alpha }\,\mathrm{\beta }\,\mathrm{\gamma }\,\mathrm{\delta }\,z\right)=\frac{\mathrm{hypergeom}\left(\left[\frac{1}{2}-\frac{\mathrm{\alpha }}{4}-\frac{\mathrm{\gamma }}{4}\right]\,\left[1-\frac{\mathrm{\alpha }}{2}\right]\,z^2\right)}{z^{\mathrm{\alpha }}}\,\mathrm{\alpha }::{\mathbb{Z}}^{-}\wedge \mathrm{\beta }=0\wedge \mathrm{\delta }=0\right]\,\left[\mathrm{HeunB}\left(\mathrm{\alpha }\,\mathrm{\beta }\,\mathrm{\gamma }\,\mathrm{\delta }\,z\right)=\mathrm{hypergeom}\left(\left[\frac{1}{2}+\frac{\mathrm{\alpha }}{4}-\frac{\mathrm{\gamma }}{4}\right]\,\left[1+\frac{\mathrm{\alpha }}{2}\right]\,z^2\right)\,\mathrm{\alpha }::\left(\neg {\mathbb{Z}}^{-}\right)\wedge \mathrm{\beta }=0\wedge \mathrm{\delta }=0\right]\right]$

$\mathrm{HeunB}\left(\mathrm{\alpha }\,\mathrm{\beta }\,2n+2+\mathrm{\alpha }\,\mathrm{\delta }\,z\right)$

$\mathrm{HeunB}\left(\mathrm{\alpha }\,\mathrm{\beta }\,2n+2+\mathrm{\alpha }\,\mathrm{\delta }\,z\right)$

$\mathrm{HB}≔\mathrm{subs}\left(n=1\,\right)$

$\mathrm{HB}≔\mathrm{HeunB}\left(\mathrm{\alpha }\,\mathrm{\beta }\,4+\mathrm{\alpha }\,\mathrm{\delta }\,z\right)$

$Q≔\mathrm{simplify}\left(\mathrm{series}\left(\mathrm{HB}\,z\,3\right)\,\mathrm{size}\right)$

$Q≔1+\frac{\mathrm{\beta }\mathrm{\alpha }+\mathrm{\beta }+\mathrm{\delta }}{2\mathrm{\alpha }+2}z+\frac{1}{8}\frac{{\mathrm{\alpha }}^2{\mathrm{\beta }}^2+\left(4{\mathrm{\beta }}^2+2\mathrm{\beta }\mathrm{\delta }-8\right)\mathrm{\alpha }+3{\mathrm{\beta }}^2+4\mathrm{\beta }\mathrm{\delta }+{\mathrm{\delta }}^2-8}{\left(\mathrm{\alpha }+1\right)\left(\mathrm{\alpha }+2\right)}{z}^2+O\left(z^3\right)$

$\mathrm{c2}≔\mathrm{coeff}\left(Q\,z\,2\right)$

$\mathrm{c2}≔\frac{{\mathrm{\alpha }}^2{\mathrm{\beta }}^2+\left(4{\mathrm{\beta }}^2+2\mathrm{\beta }\mathrm{\delta }-8\right)\mathrm{\alpha }+3{\mathrm{\beta }}^2+4\mathrm{\beta }\mathrm{\delta }+{\mathrm{\delta }}^2-8}{8\left(\mathrm{\alpha }+1\right)\left(\mathrm{\alpha }+2\right)}$

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

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

$\mathrm{\delta }=\mathrm{solve}\left(\mathrm{c2}\,\mathrm{\delta }\right)$

$\mathrm{\delta }=\mathrm{RootOf}\left({\mathrm{\_Z}}^2+\left(2\mathrm{\beta }\mathrm{\alpha }+4\mathrm{\beta }\right)\mathrm{\_Z}+{\mathrm{\alpha }}^2{\mathrm{\beta }}^2+4\mathrm{\alpha }{\mathrm{\beta }}^2+3{\mathrm{\beta }}^2-8\mathrm{\alpha }-8\right)$

$\mathrm{HB\_polynomial}≔\mathrm{subs}\left(\,\mathrm{HB}\right)$

$\mathrm{HB\_polynomial}≔\mathrm{HeunB}\left(\mathrm{\alpha }\,\mathrm{\beta }\,4+\mathrm{\alpha }\,\mathrm{RootOf}\left({\mathrm{\_Z}}^2+\left(2\mathrm{\beta }\mathrm{\alpha }+4\mathrm{\beta }\right)\mathrm{\_Z}+{\mathrm{\alpha }}^2{\mathrm{\beta }}^2+4\mathrm{\alpha }{\mathrm{\beta }}^2+3{\mathrm{\beta }}^2-8\mathrm{\alpha }-8\right)\,z\right)$

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

$1+\frac{\left(\mathrm{RootOf}\left({\mathrm{\_Z}}^2+\left(2\mathrm{\beta }\mathrm{\alpha }+4\mathrm{\beta }\right)\mathrm{\_Z}+{\mathrm{\alpha }}^2{\mathrm{\beta }}^2+4\mathrm{\alpha }{\mathrm{\beta }}^2+3{\mathrm{\beta }}^2-8\mathrm{\alpha }-8\right)+\mathrm{\beta }\mathrm{\alpha }+\mathrm{\beta }\right)z}{2\mathrm{\alpha }+2}$

$\mathrm{\_EnvExplicit}≔'\mathrm{\_EnvExplicit}'$

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

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.